sqqs1013 ch5-a122

SQQS1013 Elementary Statistics

SPECIAL SPECIAL DISTRIBUTIONSDISTRIBUTIONS

5.1 DISCRETE DISTRIBUTION

• Consist of the values a random variable can assume and the corresponding

probabilities of the values.

• The probabilities are determined theoretically or by observation.

5.1.1 Binomial Probability Distribution

• Binomial Probability Distribution – the outcomes of a binomial experiment

and the corresponding probabilities of these outcomes.

• It is applied to find the probability that an outcome will occur x times in n

performances of experiment.

• For example:

The probability of a defective laptop manufactured at a firm is 0.05 in a

random sample of ten.

The probability of 8 packages will not arrive at its destination.

• To apply the binomial probability distribution, the random variable x must be

a discrete dichotomous random variable.

• Each repetition of the experiment must result in one of two possible

outcomes.

Characteristics of a Binomial Experiment

1. Each trial can have only two outcomes or outcomes that can be reduced to

two outcomes. These outcomes can be considered as either success or

failure.

2. There must be a fixed number of trials.

3. The outcomes of each trial must be independent of each other. In other

words, the outcome of one trial does not affect the outcome of another trial.

Chapter 5: Special Distributions 1


4. The probability of success is denoted by p and that of failure by q, and p +

q=1. The probabilities p and q remain constant for each trial.

5.1.1.1 Calculating Binomial Probability

There are two methods to calculate the binomial probability:

a) Using Binomial Probability Distribution Formula

• For a binomial experiment, the probability of exactly x successes in n trials

is given by the binomial formula:

P(x) = nCx px qn-x

Where;

n = the total number of trialsp = probability of successq = 1-p = probability of failurex = number of successes in n trialsn-x = number of failures in n trials

Compute the probabilities of X successes, using the binomial formula; n= 6, X= 3, p=0.03

Solution:

a) P(x=3) = 6C3 (0.03)3(1-0.03)6-3

= 6C3 (0.03)3(0.97)3

= 0.0005

A survey found that one out five Malaysian says he or she has visited a doctor in any given month. If 10 people are selected at random, find the probability that exactly 3 will have visited a doctor last month.

Solution:


Example 1

Example 2

FORMULA

ξ∆Σ λϖβ

• The success does not mean that the corresponding outcome is considered favorable or desirable and vice versa

• The outcome to which the question refers is called a success; the outcome to which it does not refer is called a failure.


b) Using the Table of Binomial Probabilities

• The probabilities for a binomial experiment can also be read from the table of

binomial probabilities.

• For any number of trials n:

The binomial probability distribution is symmetric if p = 0.5 The binomial probability distribution is skewed to the right if p is less than

0.5 The binomial probability distribution is skewed to the left if p is greater than

0.5

Determining P (x ≤ 3) for n = 6 and p = 0.30

The table of cumulative binomial probabilities (less than)

p n x 0.10 0.15 0.20 0.25 0.30 0.35 0.40

6 0 0.5314 0.3771 0.2621 0.1780 0.1176 0.0754 0.0467

1 0.8857 0.7765 0.6554 0.5339 0.4202 0.3191 0.2333

2 0.9842 0.9527 0.9011 0.8306 0.7443 0.6471 0.5443

3 0.9987 0.9941 0.9830 0.9624 0.9295 0.8826 0.8208

4 0.9999 0.9996 0.9984 0.9954 0.9891 0.9777 0.9590

5 1.0000 1.0000 0.9999 0.9998 0.9993 0.9982 0.9959

6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

If you are using cumulative binomial probabilities table; P(X ≤ x); it is easier to calculate

various from of binomial distribution such as:

Equally P(X = x) = P(X ≤ x) - P(X ≤ x - 1) or use formulaAt most P(X ≤ x) = P(X ≤ x) (directly from the table)Less than P(X < x) = P(X ≤ x - 1) At least P(X ≥ x) = 1 – P(X ≤ x - 1)Greater than P(X > x) = 1 – P(X ≤ x)From x1 to x2 P(x1 ≤ X ≤ x2) = P(X ≤ x2) - P(X ≤ x1 - 1)


0

( ) (1 )x

n r n rr

r

P X x C p p −

=

≤ = −∑ p = 0.30

P (x ≤ 3) = 0.9295n = 6x ≤ 3

How to read the probability value from the table of binomial probabilities


Between x1 and x2 P(x1<X<x2) = P(X ≤ x2 - 1) - P(X ≤ x1)Between x1 to x2 P(x1< X ≤ x2) = P(X ≤ x2) - P(X ≤ x1 )

Compute the probability of X successes using the Binomial Table where n=2, p=0.30 and X ≤1

Solution:

P(X ≤1) = 0.9100

25% of all VCR manufactured by a large electronics company are defective. A quality control inspector randomly selects three VCRs from the production line. What is the probability that,

a) exactly one of these three VCRs is defective.

b) at least two of these VCRs are defective.

Solution:


Example 3

Example 4


EXERCISE 1

1. A burglar alarm system has 6 fail-safe components. The probability of each failing is 0.05. Find these probabilities:

a. Exactly 3 will fail

b. Less than 2 will fail

c. None will fail

2. A survey from Teenage Research Unlimited found that 30% of teenage consumers receive their spending money from part-time jobs. If 5 teenagers are selected at random, find the probability that at least 3 of them will have part-time jobs.

3. R. H Bruskin Associates Market Research found that 40% of Americans do not think that having a college education is important to succeed in the business world. If a random sample of five American is selected, find these probabilities.

a. Exactly two people will agree with that statement.

b. At most three people will agree with that statement

c. At least two people will agree with that statement

d. Fewer than three people will agree with that statement.

4. It was found that 60% of American victims of health care fraud are senior citizens. If 10 victims are randomly selected, find the probability that exactly 3 are senior citizens.

5. If 65% of the people in a community use the gym facilities in one year, find these

probabilities for a sample of 10 people

a) Exactly four people used the gym facilities.

b) At least six people not used the gym facilities

6. In a poll of 12 to 18 year old females conducted by Harris Interactive for the Gillette Company, 40% of the young female said that they expected the US to have a female president within 10 years. Supposed a random sample of 15 females from this age group selected. Find the probabilities that of young female in this sample who expect a female president within 10 years is;

a. at most 5

b. less than 4

c. at least 9

d. 6 to 9

e. in between 4 and 8



5.1.1.2 Mean and Standard Deviation of the Binomial Probability Distribution

The mean, variance and standard deviation of a binomial distribution are:

Mean = µ = np

Variance = σ2 = npq

Standard deviation = σ = npq

Where;

n = the total number of trialsp = probability of successq = 1-p = probability of failure

Find the mean, variance and standard deviation for each of the values of n and p when

the conditions for the binomial distribution are met.

a. n=100, p=0.75

Solution:

a. µ = np = 100 (0.75) = 75 σ2 = npq = 100 (0.75) (0.25) = 18.75

σ = √npq = √18.75 = 4.33

The Statistical Bulletin published by Metropolitan Life Insurance Co. reported that 2% of

all American births result in twins. If a random sample of 8000 births is taken, find the

mean, variance and standard deviation of the number of births that would result in

twins.


FORMULA

ξ∆Σ λϖβ

Example 5

Example 6


EXERCISE 2

1. It has been reported that 83% of federal government employees use e-mail. If sample of 200 federal government employees is selected, find the mean, variance and standard deviation of the number who use e-mail.

2. A survey found that 25% of Malaysian watch movie at the cinema. Find the mean, variance and standard deviation of the number of individuals who watch movie at the cinema if a random sample of 1000 Malaysian is selected at the Bintang Walk.

5.1.2 Poisson Distribution

• Derived from the French mathematician Simeon D. Poisson.

• A discrete probability distribution that is useful when n is large and p is small

and when the independent variables occur over a period of time or given

area or volume.

• Conditional to apply the Poisson Probability Distribution

x is a discrete random variable

The occurrences are random

The occurrences are independent

• The following examples show the application of the Poisson probability

distribution.

The number of accidents that occur on a highway given during a one-week

period.

The number of customers entering a grocery store during a one-hour interval.

The number of television sets sold at a department store during a given week.

The number of typing errors per page.

A certain type of fabric made contains an average 0.5 defects per 500 yards.



5.1.2.1 Calculating Poisson Probability

a) Using Poisson Probability Distribution Formula

According to the Poisson probability distribution, the probability of x occurrences in

an interval is:

P(X)=!

xe

x

λλ−

Where;

λ : mean number of occurrences in that interval

e : approximately 2.7183

Find probability P(X; λ), using the Poisson formula for P(5;4)

Solution:

P(X=5) = 4 5(4 )

5!

e−

= 0.1563

If there are 200 typographical errors randomly distributed in a 500-page manuscript, find the probability that a given page contains exactly three errors.

Solution:


FORMULA

ξ∆Σ λϖβ

Example 7

Example 8


b) Using the Table of Poisson Probabilities

• The probabilities for a Poisson distribution can also be read from the table

of Poisson probabilities.

Determining P (x ≤ 3) for λ = 1.5

The table of cumulative Poisson probabilities (less than)

λ

x 1.1 1.2 1.3 1.4 1.5 1.6 1.7

0 0.3329 0.3012 0.2725 0.2466 0.2231 0.2019 0.1827

1 0.6990 0.6626 0.6268 0.5918 0.5578 0.5249 0.4932

2 0.9004 0.8795 0.8571 0.8335 0.8088 0.7834 0.7572

3 0.9743 0.9662 0.9569 0.9463 0.9344 0.9212 0.9068

4 0.9946 0.9923 0.9893 0.9857 0.9814 0.9763 0.9704

5 0.9990 0.9985 0.9978 0.9968 0.9955 0.9940 0.9920

6 0.9999 0.9997 0.9996 0.9994 0.9991 0.9987 0.9981

7 1.0000 1.0000 0.9999 0.9999 0.9998 0.9997 0.9996

If you are using cumulative Poisson probabilities table; P(X ≤ x); it is easier to

calculate various from of binomial distribution such as:

Chapter 5: Special Distributions

Equally P(X = x) = P(X ≤ x) - P(X ≤ x - 1) or using Poisson formulaAt most P(X ≤ x) = P(X ≤ x) (directly from the table)Less than P(X < x) = P(X ≤ x - 1) At least P(X ≥ x) = 1 – P(X ≤ x - 1)Greater than P(X > x) = 1 – P(X ≤ x)From x1 to x2 P(x1 ≤ X ≤ x2) = P(X ≤ x2) - P(X ≤ x1 - 1)Between x1 and x2 P(x1<X<x2) = P(X ≤ x2 - 1) - P(X ≤ x1)Between x1 to x2 P(x1< X ≤ x2) = P(X ≤ x2) - P(X ≤ x1 )

9

How to read the probability value from the table of Poisson probabilities

λ = 1.50

( )!

kx

k

eP X x

k

λλ−

=

≤ =∑

x ≤ 3 P (x ≤ 3) = 0.9344


Find the probability P(X; λ); using Poisson table; P(10;7)

Solution:

P(X = 10) = P(X ≤ 10) – P(X ≤ 9)

= 0.9015 – 0.8305

= 0.0710

A sales firm receives on average three calls per hour on its toll-free number. For any

given hour, find the probability that it will receive the following:

a) at most 3 callsb) at least 3 callsc) five or more callsd) between 1 to 4 calls in 2 hours

Solution:

λ = 3 calls per hour

a) P(X ≤ 3) = 0.6472

b) P(X ≥ 3) = 1 – P(X ≤ 2)

= 1 – 0.4232

= 0.5768

c) P(X ≥ 5) = 1 – P(X ≤ 4)

= 1 – 0.8153

= 0.1847

λ = 6 calls in 2 hours

d) P(1< X ≤ 4) = P(X ≤ 4) – P(X ≤ 1)

= 0.2851 – 0.0174

= 0.2677


Example 9

Example 10

EXERCISE 3

1. On average a household receives 2 telemarketing phone calls per week. Using the Poisson distribution formula, find the probability that a randomly selected household receives:

a. exactly six telemarketing phone calls during a given week.

b. less than three telemarketing phone calls in one month.

2. A washing machine in a LaundryMat breaks down an average of three times per month. Using the Poisson probability distribution formula, find the probability that this machine will have

a) exactly two break downs per month

b) at most one break down per month

c) no break downs in 2 months

3. In an airport between 1900 and 2000 hours, the number of airplane that land follows a Poisson distribution with mean 0.9 per five minutes interval. Find the probability that the number of plane that land is:

a) one or less between 1900 and 1905 hours

b) more than three between 1915 and 1930 hours

4. An average of 4.8 customers comes to Malaysia Savings and Loan every hour. Find the probability that during a given hour, the number of customer will come is

a) exactly two

b) at most 2

c) none

d) more than 5 in half an hour

5. Sports Score Jay receives, on average, eight calls per hour requesting the latest sports score. The distribution is Poisson in nature. For any randomly selected hour, find the probability that the company will receive

a) at least eight calls

b) three or more calls

c) at most five calls.


5.1.2.2 Mean and Standard Deviation of the Poisson Probability Distribution

Mean, µ = λ

Variance, σ2 = λ

Standard Deviation, σ = λ

An auto salesperson sells an average of 0.9 cars per day. Find the mean, variance

and standard deviation of cars sold per day by this salesperson.

Solution:

µ = λ = 0.9 σ2 = λ = 0.9σ = √λ = √0.9 = 0.9487

An insurance salesperson sells an average of 1.4 policies per day.

a) Find the probability that this salesperson will sell no insurance policy on a

certain day.

b) Find the mean, variance and standard deviation of the probability this

salesperson will sell the policies per day.

Solution:


FORMULA

ξ∆Σ λϖβ

Example 11

Example 12

5.2 CONTINUOUS DISTRIBUTION

• In Chapter 4, we defined a continuous random variable as a random variable

whose values are not countable.

• A continuous random variable can assume any value over an interval or

intervals because the number of values contained in any interval is infinite.

• The possible number of values that a continuous random variable can

assume is also infinite.

• Example of continuous random variables:

The life of battery, heights of people, time taken to complete an examination,

amount of milk in a gallon, weights of babies, prices of houses.

• The probability distribution of continuous random variable has two

characteristics:

1. The probability that x assumes a value in any interval lies in the range 0 to 1.

Figure 1: Area under a curve between two points.

2. The total probability of all the (mutually exclusive) intervals within which x can

assume a value is 1.0.

Figure 2: Total area under a probability distribution.


5.2.1 Normal Distribution

• The normal distribution is the most important and most commonly used

among all of probability distributions.

• A large number of phenomena in the real world are normally distributed

either exactly or approximately.

• The normal probability distribution or the normal curve is a bell-shaped

(symmetric) curve.

• Its mean is denoted by µ while its standard deviation is denoted by σ.

Figure 3: Normal distribution with mean; µ and standard deviation; σ .

• A plotted normal probability distribution will gives a bell-shaped curve which

can be illustrated likes:

a) The total area under the curve is 1.0. b) The curve is symmetric about the

mean.

c)


c) The two tails of the curve extend indefinitely which means that the curve never

touch x- axis.

5.2.1.1 The Standard Normal Distribution

• Is a normal distribution with µ = 0 and σ = 1.

• The value under the curve indicates the proportion of area in each section.

(example figure 2; pg 13)

• The units for the standard normal distribution curve are denoted by z and

called the z values or z scores.

• The z value or z score is actually the number of standard deviation that a

particular x value is away from the mean.

• The area under a standard normal distribution curve is used to solve

practical application problems such as:

finding the percentage of adult woman whose height is between 5 feet 4

inches and 5 feet 7 inches.


A. Finding areas under the standard normal distribution curve

• The standard normal distribution table lists the areas under the

standard normal curve to the left of z-values from –3.49 to 3.49.

• Although the z-values on the left side of the mean are negative, the area

under the curve is always positive.

Determine the area under the standard normal curve to the left of z = 1.95


How to read the probability value from the standard normal distribution table

Note: z = 1.95 can be interpreted as area to the left of 1.95. P(z < 1.95) = 0.9744 or P(z < 1.95) = P(z ≤ 1.95) = 0.9744The probability that a continuous random variable assumes a single value is zero. Therefore, P(z = 1.95) = 0.

Find the area under the standard normal curve:a) To the left of z = 1.56b) To the right of z = -1.32c) From z = 0.85 to z = 1.95

Solution:

a) To the left of z = 1.56

P(z < 1.56) = 0.9406

b) To the right of z = -1.32

P(z > -1.32) = 1 – P(z < -1.32)

= 1 - 0.0934

= 0.9066

c) From z = 0.85 to z = 1.95

P(0.85 ≤ z ≤ 1.95) = P(z ≤ 1.95) – P(z ≤ 0.85)


Example 13

0 1.56

-1.32 0

0.85 1.95

= 0.9744 – 0.8023

= 0.1721

EXERCISE 4

Find the area under the standard normal curve:

a) To the left of z = -2.87

b) To the right of z = 2.45

c) Between z = -2.15 and z=1.67

B. Converting an x Value to a z Value

For a normal variable x, a particular value of x can be converted to its

corresponding z value by using the formula:

z = x µ

σ−

Where µ and σ are the mean and standard deviation of the normal distribution of

x, respectively.


FORMULA

ξ∆Σ λϖβ

Remember!

The z value for the mean of a normal distribution is always zero.

Let x be a continuous random variable that has a normal distribution with a mean of 50

and a standard deviation of 10. Convert the following x values to z values.

a) 55 b) 35

Solution:

For the given normal distribution, µ = 50 and σ =10

a) The z value for x = 55 is computes as follows:

b) z = 35 – 50 = 10

Let x be a continuous random variable that is normally distributed with a mean of 65 and

a standard deviation of 15. Find the probability that x can assumes a value:

a) less than 43

b) greater than 74

c) between 56 and 71

Solution:


Example 14

Example 15

EXERCISE 5

1. Let x denote the time takes to run a road race. Supposed x is approximately normally distributed with mean of 190 minutes and standard deviation of 21 minutes. If one runner is selected at random, what is the probability that this runner will complete this road race?

a) in less than 150 minutes

b) in 205 to 245 minutes

2. The mean number of hours a student spends on the computer is 3.1 hours per day. Assume the standard deviation is 0.5 hour. Find the percentage of students who spend less than 3.5 hours on the computer. Assume the variable is normally distributed.

3. The score of 6000 candidates in a certain examination are found to be approximately normal distributed with a mean of 55 and a standard deviation of 10:

a) If a score of 75 or more is required for passing the distinction, estimate the

number of grades with distinction.

b) Calculate the probability that a candidate selected at random has a score

between 45 and 65.

5.3 INTRODUCTION TO t- DISTRIBUTION

• The t distribution is very similar to the standardized normal distribution.

• Both distributions are bell-shaped and symmetrical.

• However, the t distribution has more area in the tails and less in the center

than does the standardized normal distribution.

• This is because σ is unknown and S is used to estimate it.

• Because the value of σ is uncertain, the values of t that are observed will be

more variable than for Z.


Standard normalt distribution for 5 degrees of freedom

• As the number of degrees of freedom increases, the t distribution gradually

approaches the standardized normal distribution until the two are virtually

identical.

• This happens because S becomes a better estimate of σ as the sample size

gets larger.

• With a sample size of about 120 or more, S estimates σ precisely enough

that there is little difference between the t and Z distributions.

• For this reason, most statisticians use Z instead of t when the sample size is

greater than 120.

EXERCISE 6

1. 10 % of the bulbs produced by a factory are defective. A sample of 5 bulbs is selected randomly and tested for defect. Find the probability that

a) two bulbs are defective

b) at least one bulb is defective

2. In a university, 20 percent of the students fail the statistic test. If 20 students from the

university are interviewed, what is the probability of getting:

a) less than 3 students who fail the test

b) exactly 4 students who fail the test

3. A financial institution in Kuala Lumpur has offer a job as a risk analyst. For the minimum qualification, the applicant must seats for writing test. Based on the management experience, 40% of the applicants will pass the test and qualified for the interview session. There are 20 applicants who have applied for the jobs. Find the probability that there are more than 50% applicants will pass the test.


4. An Elementary Statistic class has 75 members. If there is a 12% absentee rate per class meeting, find the mean, variance and standard deviation of the number of students who will be absent from each class.

5. Before an umbrella leaves the factory, it is given a quality control check. The probability that an umbrella contains zero, one or two defects is 0.88, 0.08 and 0.04 respectively. In a sample of 16 umbrellas, find the probability that:

a) 9 will have no defect

b) 4 will have one defect

c) 3 will have two defect

6. En. Rostam is a credit officer at the Trust Bank. Based on his experience, he estimates that an average he will receive the loan application in a week is 3 applications. Find the probability that:

a) he receives none loan application in a week

b) he receives 2 until 5 loan applications in a week

c) at least 5 loan applications he receives in 14 days.

7. A bookstore owner examines 5 books from each lot of 25 to check for missing pages. If he finds at least two books with missing pages, the entire lot is returned. If indeed, there are five books with missing pages, find the probability that the lot will be returned.

8. The numbers of customers who enter shop ABC independent of one another and at random intervals follow a Poisson distribution with an average rate 42 customers per hour. Find the probability that:

a) no customer enter the shop during a particular 1 minutes interval

b) at least 4 customers enter the shop during a particular 5 minutes interval

c) between 2 and 6 customers enter the shop during a particular 10-minute

interval.

9. One research has been conducted by Student’s Affair Department of Menara University about the CGPA of final semester student for 2010/2011 session. The outcome of the research showed that the CGPA of the student is normally distributed with mean 2.80 and standard deviation of 0.40. If one final semester student has been selected at random;

a) Calculate the probability that the student’s CGPA is between 2.00 and 3.00

b) Find the percentage that the student’s CGPA is less than 2.00

c) Calculate the probability that the student’s CGPA is at least 3.00


d) Calculate the probability that the student’s CGPA is more than 3.70 (first

class honors). If the university has 1000 final semester students, find the

number of first class honors student.

10. Encik Ahmad works as a lawyer at his own law firm which situated at Bandar Kenangan. He drives his car to go to his workplace and back everyday. The estimated time taken for the journey is normal distributed with mean 24 minutes and standard deviation is 3.8 minutes.

a) Calculate the probability of estimated time taken by Encik Ahmad to go to

his workplace and back is at least half an hour.

b) Find the probability of estimated time taken by Encik Ahmad to go to his

workplace and back is 20 minutes to 25 minutes

c) Find the percentage that the estimated time taken by Encik Ahmad to go

to his workplace and back is more than 25 minutes.

d) Find the probability of estimated time taken by Encik Ahmad to go to his

workplace and back is less than 10 minutes. Explain.

11. Assume X is timing for a runner to finish his 2 km run. Given X is normally distributed with mean 15 minutes and standard deviation is 3 minutes. If one runner is selected at random, find the probability that the runner can finish his 2km run in time;

a) Less than 13 minutes

b) Not more than 16 minutes

c) Within 14 minutes and 17 minutes.

12. Given the systolic blood pressure for the obesity group has mean 132 mmHg and standard deviation 8 mmHg. Assume the variable normally distributed, find the probability an obese person that has been selected at random have a systolic blood pressure:

a) More than 130 mmHg.

b) Less than 140 mmHg.

c) Between 131 mmHg and 136 mmHg.

13. The number of passenger for domestic flight from Alor Setar to Kuala Lumpur is normally distributed with mean 80 and standard deviation 12. If one domestic flight is selected at random, find the probability the plain has a passenger:


a) less than 90 passengers

b) at least 75 passengers

c) between 79 to 95 passengers


BINOMIAL DISTRIBUTION

p n r 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

1 0 0.9000 0.8500 0.8000 0.7500 0.7000 0.6500 0.6000 0.5500 0.5000

1 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

2 0 0.8100 0.7225 0.6400 0.5625 0.4900 0.4225 0.3600 0.3025 0.2500

1 0.9900 0.9775 0.9600 0.9375 0.9100 0.8775 0.8400 0.7975 0.7500

2 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

3 0 0.7290 0.6141 0.5120 0.4219 0.3430 0.2746 0.2160 0.1664 0.1250

1 0.9720 0.9393 0.8960 0.8438 0.7840 0.7183 0.6480 0.5748 0.5000

2 0.9990 0.9966 0.9920 0.9844 0.9730 0.9571 0.9360 0.9089 0.8750

3 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

4 0 0.6561 0.5220 0.4096 0.3164 0.2401 0.1785 0.1296 0.0915 0.0625

1 0.9477 0.8905 0.8192 0.7383 0.6517 0.5630 0.4752 0.3910 0.3125

2 0.9963 0.9880 0.9728 0.9492 0.9163 0.8735 0.8208 0.7585 0.6875

3 0.9999 0.9995 0.9984 0.9961 0.9919 0.9850 0.9744 0.9590 0.9375

4 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

5 0 0.5905 0.4437 0.3277 0.2373 0.1681 0.1160 0.0778 0.0503 0.0313

1 0.9185 0.8352 0.7373 0.6328 0.5282 0.4284 0.3370 0.2562 0.1875

2 0.9914 0.9734 0.9421 0.8965 0.8369 0.7648 0.6826 0.5931 0.5000

3 0.9995 0.9978 0.9933 0.9844 0.9692 0.9460 0.9130 0.8688 0.8125

4 1.0000 0.9999 0.9997 0.9990 0.9976 0.9947 0.9898 0.9815 0.9688

5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

6 0 0.5314 0.3771 0.2621 0.1780 0.1176 0.0754 0.0467 0.0277 0.0156

1 0.8857 0.7765 0.6554 0.5339 0.4202 0.3191 0.2333 0.1636 0.1094

2 0.9842 0.9527 0.9011 0.8306 0.7443 0.6471 0.5443 0.4415 0.3438

3 0.9987 0.9941 0.9830 0.9624 0.9295 0.8826 0.8208 0.7447 0.6563

4 0.9999 0.9996 0.9984 0.9954 0.9891 0.9777 0.9590 0.9308 0.8906

5 1.0000 1.0000 0.9999 0.9998 0.9993 0.9982 0.9959 0.9917 0.9844

6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

7 0 0.4783 0.3206 0.2097 0.1335 0.0824 0.0490 0.0280 0.0152 0.0078

1 0.8503 0.7166 0.5767 0.4449 0.3294 0.2338 0.1586 0.1024 0.0625

2 0.9743 0.9262 0.8520 0.7564 0.6471 0.5323 0.4199 0.3164 0.2266

3 0.9973 0.9879 0.9667 0.9294 0.8740 0.8002 0.7102 0.6083 0.5000

4 0.9998 0.9988 0.9953 0.9871 0.9712 0.9444 0.9037 0.8471 0.7734

5 1.0000 0.9999 0.9996 0.9987 0.9962 0.9910 0.9812 0.9643 0.9375

6 1.0000 1.0000 1.0000 0.9999 0.9998 0.9994 0.9984 0.9963 0.9922

7 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

continue;Chapter 5: Special DistributionBinomial Distribution Table

25

0

( ) (1 )xn r n r

rr

P X x C p p −

=

≤ = −∑

p

Chapter 5: Special DistributionBinomial Distribution Table

26

n r 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

8 0 0.4305 0.2725 0.1678 0.1001 0.0576 0.0319 0.0168 0.0084 0.0039

1 0.8131 0.6572 0.5033 0.3671 0.2553 0.1691 0.1064 0.0632 0.0352

2 0.9619 0.8948 0.7969 0.6785 0.5518 0.4278 0.3154 0.2201 0.1445

3 0.9950 0.9786 0.9437 0.8862 0.8059 0.7064 0.5941 0.4770 0.3633

4 0.9996 0.9971 0.9896 0.9727 0.9420 0.8939 0.8263 0.7396 0.6367

5 1.0000 0.9998 0.9988 0.9958 0.9887 0.9747 0.9502 0.9115 0.8555

6 1.0000 1.0000 0.9999 0.9996 0.9987 0.9964 0.9915 0.9819 0.9648

7 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9993 0.9983 0.9961

8 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

9 0 0.3874 0.2316 0.1342 0.0751 0.0404 0.0207 0.0101 0.0046 0.0020

1 0.7748 0.5995 0.4362 0.3003 0.1960 0.1211 0.0705 0.0385 0.0195

2 0.9470 0.8591 0.7382 0.6007 0.4628 0.3373 0.2318 0.1495 0.0898

3 0.9917 0.9661 0.9144 0.8343 0.7297 0.6089 0.4826 0.3614 0.2539

4 0.9991 0.9944 0.9804 0.9511 0.9012 0.8283 0.7334 0.6214 0.5000

5 0.9999 0.9994 0.9969 0.9900 0.9747 0.9464 0.9006 0.8342 0.7461

6 1.0000 1.0000 0.9997 0.9987 0.9957 0.9888 0.9750 0.9502 0.9102

7 1.0000 1.0000 1.0000 0.9999 0.9996 0.9986 0.9962 0.9909 0.9805

8 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9992 0.9980

9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

10 0 0.3487 0.1969 0.1074 0.0563 0.0282 0.0135 0.0060 0.0025 0.0010

1 0.7361 0.5443 0.3758 0.2440 0.1493 0.0860 0.0464 0.0233 0.0107

2 0.9298 0.8202 0.6778 0.5256 0.3828 0.2616 0.1673 0.0996 0.0547

3 0.9872 0.9500 0.8791 0.7759 0.6496 0.5138 0.3823 0.2660 0.1719

4 0.9984 0.9901 0.9672 0.9219 0.8497 0.7515 0.6331 0.5044 0.3770

5 0.9999 0.9986 0.9936 0.9803 0.9527 0.9051 0.8338 0.7384 0.6230

6 1.0000 0.9999 0.9991 0.9965 0.9894 0.9740 0.9452 0.8980 0.8281

7 1.0000 1.0000 0.9999 0.9996 0.9984 0.9952 0.9877 0.9726 0.9453

8 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 0.9983 0.9955 0.9893

9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9990

10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

11 0 0.3138 0.1673 0.0859 0.0422 0.0198 0.0088 0.0036 0.0014 0.0005

1 0.6974 0.4922 0.3221 0.1971 0.1130 0.0606 0.0302 0.0139 0.0059

2 0.9104 0.7788 0.6174 0.4552 0.3127 0.2001 0.1189 0.0652 0.0327

3 0.9815 0.9306 0.8389 0.7133 0.5696 0.4256 0.2963 0.1911 0.1133

4 0.9972 0.9841 0.9496 0.8854 0.7897 0.6683 0.5328 0.3971 0.2744

5 0.9997 0.9973 0.9883 0.9657 0.9218 0.8513 0.7535 0.6331 0.5000

6 1.0000 0.9997 0.9980 0.9924 0.9784 0.9499 0.9006 0.8262 0.7256

7 1.0000 1.0000 0.9998 0.9988 0.9957 0.9878 0.9707 0.9390 0.8867

8 1.0000 1.0000 1.0000 0.9999 0.9994 0.9980 0.9941 0.9852 0.9673

9 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9993 0.9978 0.9941

10 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9995

11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000Continue;

p


27

n r 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

12 0 0.2824 0.1422 0.0687 0.0317 0.0138 0.0057 0.0022 0.0008 0.0002

1 0.6590 0.4435 0.2749 0.1584 0.0850 0.0424 0.0196 0.0083 0.0032

2 0.8891 0.7358 0.5583 0.3907 0.2528 0.1513 0.0834 0.0421 0.0193

3 0.9744 0.9078 0.7946 0.6488 0.4925 0.3467 0.2253 0.1345 0.0730

4 0.9957 0.9761 0.9274 0.8424 0.7237 0.5833 0.4382 0.3044 0.1938

5 0.9995 0.9954 0.9806 0.9456 0.8822 0.7873 0.6652 0.5269 0.3872

6 0.9999 0.9993 0.9961 0.9857 0.9614 0.9154 0.8418 0.7393 0.6128

7 1.0000 0.9999 0.9994 0.9972 0.9905 0.9745 0.9427 0.8883 0.8062

8 1.0000 1.0000 0.9999 0.9996 0.9983 0.9944 0.9847 0.9644 0.9270

9 1.0000 1.0000 1.0000 1.0000 0.9998 0.9992 0.9972 0.9921 0.9807

10 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9989 0.9968

11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998

12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

13 0 0.2542 0.1209 0.0550 0.0238 0.0097 0.0037 0.0013 0.0004 0.0001

1 0.6213 0.3983 0.2336 0.1267 0.0637 0.0296 0.0126 0.0049 0.0017

2 0.8661 0.6920 0.5017 0.3326 0.2025 0.1132 0.0579 0.0269 0.0112

3 0.9658 0.8820 0.7473 0.5843 0.4206 0.2783 0.1686 0.0929 0.0461

4 0.9935 0.9658 0.9009 0.7940 0.6543 0.5005 0.3530 0.2279 0.1334

5 0.9991 0.9925 0.9700 0.9198 0.8346 0.7159 0.5744 0.4268 0.2905

6 0.9999 0.9987 0.9930 0.9757 0.9376 0.8705 0.7712 0.6437 0.5000

7 1.0000 0.9998 0.9988 0.9944 0.9818 0.9538 0.9023 0.8212 0.7095

8 1.0000 1.0000 0.9998 0.9990 0.9960 0.9874 0.9679 0.9302 0.8666

9 1.0000 1.0000 1.0000 0.9999 0.9993 0.9975 0.9922 0.9797 0.9539

10 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9987 0.9959 0.9888

11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 0.9983

12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999

13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

14 0 0.2288 0.1028 0.0440 0.0178 0.0068 0.0024 0.0008 0.0002 0.0001

1 0.5846 0.3567 0.1979 0.1010 0.0475 0.0205 0.0081 0.0029 0.0009

2 0.8416 0.6479 0.4481 0.2811 0.1608 0.0839 0.0398 0.0170 0.0065

3 0.9559 0.8535 0.6982 0.5213 0.3552 0.2205 0.1243 0.0632 0.0287

4 0.9908 0.9533 0.8702 0.7415 0.5842 0.4227 0.2793 0.1672 0.0898

5 0.9985 0.9885 0.9561 0.8883 0.7805 0.6405 0.4859 0.3373 0.2120

6 0.9998 0.9978 0.9884 0.9617 0.9067 0.8164 0.6925 0.5461 0.3953

7 1.0000 0.9997 0.9976 0.9897 0.9685 0.9247 0.8499 0.7414 0.6047

8 1.0000 1.0000 0.9996 0.9978 0.9917 0.9757 0.9417 0.8811 0.7880

9 1.0000 1.0000 1.0000 0.9997 0.9983 0.9940 0.9825 0.9574 0.9102

10 1.0000 1.0000 1.0000 1.0000 0.9998 0.9989 0.9961 0.9886 0.9713

11 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9994 0.9978 0.9935

12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9991

13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999

14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

Continue; p


28

n r 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

15 0 0.2059 0.0874 0.0352 0.0134 0.0047 0.0016 0.0005 0.0001 0.0000

1 0.5490 0.3186 0.1671 0.0802 0.0353 0.0142 0.0052 0.0017 0.0005

2 0.8159 0.6042 0.3980 0.2361 0.1268 0.0617 0.0271 0.0107 0.0037

3 0.9444 0.8227 0.6482 0.4613 0.2969 0.1727 0.0905 0.0424 0.0176

4 0.9873 0.9383 0.8358 0.6865 0.5155 0.3519 0.2173 0.1204 0.0592

5 0.9978 0.9832 0.9389 0.8516 0.7216 0.5643 0.4032 0.2608 0.1509

6 0.9997 0.9964 0.9819 0.9434 0.8689 0.7548 0.6098 0.4522 0.3036

7 1.0000 0.9994 0.9958 0.9827 0.9500 0.8868 0.7869 0.6535 0.5000

8 1.0000 0.9999 0.9992 0.9958 0.9848 0.9578 0.9050 0.8182 0.6964

9 1.0000 1.0000 0.9999 0.9992 0.9963 0.9876 0.9662 0.9231 0.8491

10 1.0000 1.0000 1.0000 0.9999 0.9993 0.9972 0.9907 0.9745 0.9408

11 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 0.9981 0.9937 0.9824

12 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9989 0.9963

13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995

14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

16 0 0.1853 0.0743 0.0281 0.0100 0.0033 0.0010 0.0003 0.0001 0.0000

1 0.5147 0.2839 0.1407 0.0635 0.0261 0.0098 0.0033 0.0010 0.0003

2 0.7892 0.5614 0.3518 0.1971 0.0994 0.0451 0.0183 0.0066 0.0021

3 0.9316 0.7899 0.5981 0.4050 0.2459 0.1339 0.0651 0.0281 0.0106

4 0.9830 0.9209 0.7982 0.6302 0.4499 0.2892 0.1666 0.0853 0.0384

5 0.9967 0.9765 0.9183 0.8103 0.6598 0.4900 0.3288 0.1976 0.1051

6 0.9995 0.9944 0.9733 0.9204 0.8247 0.6881 0.5272 0.3660 0.2272

7 0.9999 0.9989 0.9930 0.9729 0.9256 0.8406 0.7161 0.5629 0.4018

8 1.0000 0.9998 0.9985 0.9925 0.9743 0.9329 0.8577 0.7441 0.5982

9 1.0000 1.0000 0.9998 0.9984 0.9929 0.9771 0.9417 0.8759 0.7728

10 1.0000 1.0000 1.0000 0.9997 0.9984 0.9938 0.9809 0.9514 0.8949

11 1.0000 1.0000 1.0000 1.0000 0.9997 0.9987 0.9951 0.9851 0.9616

12 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9991 0.9965 0.9894

13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9994 0.9979

14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997

15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

17 0 0.1668 0.0631 0.0225 0.0075 0.0023 0.0007 0.0002 0.0000 0.0000

1 0.4818 0.2525 0.1182 0.0501 0.0193 0.0067 0.0021 0.0006 0.0001

2 0.7618 0.5198 0.3096 0.1637 0.0774 0.0327 0.0123 0.0041 0.0012

3 0.9174 0.7556 0.5489 0.3530 0.2019 0.1028 0.0464 0.0184 0.0064

4 0.9779 0.9013 0.7582 0.5739 0.3887 0.2348 0.1260 0.0596 0.0245

5 0.9953 0.9681 0.8943 0.7653 0.5968 0.4197 0.2639 0.1471 0.0717

6 0.9992 0.9917 0.9623 0.8929 0.7752 0.6188 0.4478 0.2902 0.1662

7 0.9999 0.9983 0.9891 0.9598 0.8954 0.7872 0.6405 0.4743 0.3145

8 1.0000 0.9997 0.9974 0.9876 0.9597 0.9006 0.8011 0.6626 0.5000Continue;

p


29

n r 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

9 1.0000 1.0000 0.9995 0.9969 0.9873 0.9617 0.9081 0.8166 0.6855

10 1.0000 1.0000 0.9999 0.9994 0.9968 0.9880 0.9652 0.9174 0.8338

11 1.0000 1.0000 1.0000 0.9999 0.9993 0.9970 0.9894 0.9699 0.9283

12 1.0000 1.0000 1.0000 1.0000 0.9999 0.9994 0.9975 0.9914 0.9755

13 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 0.9981 0.9936

14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9988

15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999

16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

18 0 0.1501 0.0536 0.0180 0.0056 0.0016 0.0004 0.0001 0.0000 0.0000

1 0.4503 0.2241 0.0991 0.0395 0.0142 0.0046 0.0013 0.0003 0.0001

2 0.7338 0.4797 0.2713 0.1353 0.0600 0.0236 0.0082 0.0025 0.0007

3 0.9018 0.7202 0.5010 0.3057 0.1646 0.0783 0.0328 0.0120 0.0038

4 0.9718 0.8794 0.7164 0.5187 0.3327 0.1886 0.0942 0.0411 0.0154

5 0.9936 0.9581 0.8671 0.7175 0.5344 0.3550 0.2088 0.1077 0.0481

6 0.9988 0.9882 0.9487 0.8610 0.7217 0.5491 0.3743 0.2258 0.1189

7 0.9998 0.9973 0.9837 0.9431 0.8593 0.7283 0.5634 0.3915 0.2403

8 1.0000 0.9995 0.9957 0.9807 0.9404 0.8609 0.7368 0.5778 0.4073

9 1.0000 0.9999 0.9991 0.9946 0.9790 0.9403 0.8653 0.7473 0.5927

10 1.0000 1.0000 0.9998 0.9988 0.9939 0.9788 0.9424 0.8720 0.7597

11 1.0000 1.0000 1.0000 0.9998 0.9986 0.9938 0.9797 0.9463 0.8811

12 1.0000 1.0000 1.0000 1.0000 0.9997 0.9986 0.9942 0.9817 0.9519

13 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9987 0.9951 0.9846

14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998 0.9990 0.9962

15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9993

16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999

17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

18 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

19 0 0.1351 0.0456 0.0144 0.0042 0.0011 0.0003 0.0001 0.0000 0.0000

1 0.4203 0.1985 0.0829 0.0310 0.0104 0.0031 0.0008 0.0002 0.0000

2 0.7054 0.4413 0.2369 0.1113 0.0462 0.0170 0.0055 0.0015 0.0004

3 0.8850 0.6841 0.4551 0.2631 0.1332 0.0591 0.0230 0.0077 0.0022

4 0.9648 0.8556 0.6733 0.4654 0.2822 0.1500 0.0696 0.0280 0.0096

5 0.9914 0.9463 0.8369 0.6678 0.4739 0.2968 0.1629 0.0777 0.0318

6 0.9983 0.9837 0.9324 0.8251 0.6655 0.4812 0.3081 0.1727 0.0835

7 0.9997 0.9959 0.9767 0.9225 0.8180 0.6656 0.4878 0.3169 0.1796

8 1.0000 0.9992 0.9933 0.9713 0.9161 0.8145 0.6675 0.4940 0.3238

9 1.0000 0.9999 0.9984 0.9911 0.9674 0.9125 0.8139 0.6710 0.5000

10 1.0000 1.0000 0.9997 0.9977 0.9895 0.9653 0.9115 0.8159 0.6762

11 1.0000 1.0000 1.0000 0.9995 0.9972 0.9886 0.9648 0.9129 0.8204

12 1.0000 1.0000 1.0000 0.9999 0.9994 0.9969 0.9884 0.9658 0.9165

13 1.0000 1.0000 1.0000 1.0000 0.9999 0.9993 0.9969 0.9891 0.9682Continue:

p


30

n r 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

14 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9994 0.9972 0.9904

15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9995 0.9978

16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996

17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

18 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

19 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

20 0 0.1216 0.0388 0.0115 0.0032 0.0008 0.0002 0.0000 0.0000 0.0000

1 0.3917 0.1756 0.0692 0.0243 0.0076 0.0021 0.0005 0.0001 0.0000

2 0.6769 0.4049 0.2061 0.0913 0.0355 0.0121 0.0036 0.0009 0.0002

3 0.8670 0.6477 0.4114 0.2252 0.1071 0.0444 0.0160 0.0049 0.0013

4 0.9568 0.8298 0.6296 0.4148 0.2375 0.1182 0.0510 0.0189 0.0059

5 0.9887 0.9327 0.8042 0.6172 0.4164 0.2454 0.1256 0.0553 0.0207

6 0.9976 0.9781 0.9133 0.7858 0.6080 0.4166 0.2500 0.1299 0.0577

7 0.9996 0.9941 0.9679 0.8982 0.7723 0.6010 0.4159 0.2520 0.1316

8 0.9999 0.9987 0.9900 0.9591 0.8867 0.7624 0.5956 0.4143 0.2517

9 1.0000 0.9998 0.9974 0.9861 0.9520 0.8782 0.7553 0.5914 0.4119

10 1.0000 1.0000 0.9994 0.9961 0.9829 0.9468 0.8725 0.7507 0.5881

11 1.0000 1.0000 0.9999 0.9991 0.9949 0.9804 0.9435 0.8692 0.7483

12 1.0000 1.0000 1.0000 0.9998 0.9987 0.9940 0.9790 0.9420 0.8684

13 1.0000 1.0000 1.0000 1.0000 0.9997 0.9985 0.9935 0.9786 0.9423

14 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9984 0.9936 0.9793

15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9985 0.9941

16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9997 0.9987

17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9998

18 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

19 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

20 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000


31

POISSON DISTRIBUTION

λ

x 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

0 0.9048 0.8187 0.7408 0.6703 0.6065 0.5488 0.4966 0.4493 0.4066 0.3679

1 0.9953 0.9825 0.9631 0.9384 0.9098 0.8781 0.8442 0.8088 0.7725 0.7358

2 0.9998 0.9989 0.9964 0.9921 0.9856 0.9769 0.9659 0.9526 0.9371 0.9197

3 1.0000 0.9999 0.9997 0.9992 0.9982 0.9966 0.9942 0.9909 0.9865 0.9810

4 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996 0.9992 0.9986 0.9977 0.9963

5 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9997 0.9994

6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999

7 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

λ

x 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0

0 0.3329 0.3012 0.2725 0.2466 0.2231 0.2019 0.1827 0.1653 0.1496 0.1353

1 0.6990 0.6626 0.6268 0.5918 0.5578 0.5249 0.4932 0.4628 0.4337 0.4060

2 0.9004 0.8795 0.8571 0.8335 0.8088 0.7834 0.7572 0.7306 0.7037 0.6767

3 0.9743 0.9662 0.9569 0.9463 0.9344 0.9212 0.9068 0.8913 0.8747 0.8571

4 0.9946 0.9923 0.9893 0.9857 0.9814 0.9763 0.9704 0.9636 0.9559 0.9473

5 0.9990 0.9985 0.9978 0.9968 0.9955 0.9940 0.9920 0.9896 0.9868 0.9834

6 0.9999 0.9997 0.9996 0.9994 0.9991 0.9987 0.9981 0.9974 0.9966 0.9955

7 1.0000 1.0000 0.9999 0.9999 0.9998 0.9997 0.9996 0.9994 0.9992 0.9989

8 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9998 0.9998

9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

λ

x 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0

0 0.1225 0.1108 0.1003 0.0907 0.0821 0.0743 0.0672 0.0608 0.0550 0.0498

1 0.3796 0.3546 0.3309 0.3084 0.2873 0.2674 0.2487 0.2311 0.2146 0.1991

2 0.6496 0.6227 0.5960 0.5697 0.5438 0.5184 0.4936 0.4695 0.4460 0.4232

3 0.8386 0.8194 0.7993 0.7787 0.7576 0.7360 0.7141 0.6919 0.6696 0.6472

4 0.9379 0.9275 0.9162 0.9041 0.8912 0.8774 0.8629 0.8477 0.8318 0.8153

5 0.9796 0.9751 0.9700 0.9643 0.9580 0.9510 0.9433 0.9349 0.9258 0.9161

6 0.9941 0.9925 0.9906 0.9884 0.9858 0.9828 0.9794 0.9756 0.9713 0.9665

7 0.9985 0.9980 0.9974 0.9967 0.9958 0.9947 0.9934 0.9919 0.9901 0.9881

8 0.9997 0.9995 0.9994 0.9991 0.9989 0.9985 0.9981 0.9976 0.9969 0.9962

9 0.9999 0.9999 0.9999 0.9998 0.9997 0.9996 0.9995 0.9993 0.9991 0.9989

10 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9998 0.9998 0.9997

11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999

12 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

λ Chapter 5: Special DistributionPoisson Distribution Table

32

0

( )!

kx

k

eP X x

k

λλ−

=

≤ =∑

x 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 4.0

0 0.0450 0.0408 0.0369 0.0334 0.0302 0.0273 0.0247 0.0224 0.0202 0.0183

1 0.1847 0.1712 0.1586 0.1468 0.1359 0.1257 0.1162 0.1074 0.0992 0.0916

2 0.4012 0.3799 0.3594 0.3397 0.3208 0.3027 0.2854 0.2689 0.2531 0.2381

3 0.6248 0.6025 0.5803 0.5584 0.5366 0.5152 0.4942 0.4735 0.4532 0.4335

4 0.7982 0.7806 0.7626 0.7442 0.7254 0.7064 0.6872 0.6678 0.6484 0.6288

5 0.9057 0.8946 0.8829 0.8705 0.8576 0.8441 0.8301 0.8156 0.8006 0.7851

6 0.9612 0.9554 0.9490 0.9421 0.9347 0.9267 0.9182 0.9091 0.8995 0.8893

7 0.9858 0.9832 0.9802 0.9769 0.9733 0.9692 0.9648 0.9599 0.9546 0.9489

8 0.9953 0.9943 0.9931 0.9917 0.9901 0.9883 0.9863 0.9840 0.9815 0.9786

9 0.9986 0.9982 0.9978 0.9973 0.9967 0.9960 0.9952 0.9942 0.9931 0.9919

10 0.9996 0.9995 0.9994 0.9992 0.9990 0.9987 0.9984 0.9981 0.9977 0.9972

11 0.9999 0.9999 0.9998 0.9998 0.9997 0.9996 0.9995 0.9994 0.9993 0.9991

12 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998 0.9997

13 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999

14 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

λ

x 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 5.0

0 0.0166 0.0150 0.0136 0.0123 0.0111 0.0101 0.0091 0.0082 0.0074 0.0067

1 0.0845 0.0780 0.0719 0.0663 0.0611 0.0563 0.0518 0.0477 0.0439 0.0404

2 0.2238 0.2102 0.1974 0.1851 0.1736 0.1626 0.1523 0.1425 0.1333 0.1247

3 0.4142 0.3954 0.3772 0.3594 0.3423 0.3257 0.3097 0.2942 0.2793 0.2650

4 0.6093 0.5898 0.5704 0.5512 0.5321 0.5132 0.4946 0.4763 0.4582 0.4405

5 0.7693 0.7531 0.7367 0.7199 0.7029 0.6858 0.6684 0.6510 0.6335 0.6160

6 0.8786 0.8675 0.8558 0.8436 0.8311 0.8180 0.8046 0.7908 0.7767 0.7622

7 0.9427 0.9361 0.9290 0.9214 0.9134 0.9049 0.8960 0.8867 0.8769 0.8666

8 0.9755 0.9721 0.9683 0.9642 0.9597 0.9549 0.9497 0.9442 0.9382 0.9319

9 0.9905 0.9889 0.9871 0.9851 0.9829 0.9805 0.9778 0.9749 0.9717 0.9682

10 0.9966 0.9959 0.9952 0.9943 0.9933 0.9922 0.9910 0.9896 0.9880 0.9863

11 0.9989 0.9986 0.9983 0.9980 0.9976 0.9971 0.9966 0.9960 0.9953 0.9945

12 0.9997 0.9996 0.9995 0.9993 0.9992 0.9990 0.9988 0.9986 0.9983 0.9980

13 0.9999 0.9999 0.9998 0.9998 0.9997 0.9997 0.9996 0.9995 0.9994 0.9993

14 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998

15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999

16 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

λ

x 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0

0 0.0061 0.0055 0.0050 0.0045 0.0041 0.0037 0.0033 0.0030 0.0027 0.0025

1 0.0372 0.0342 0.0314 0.0289 0.0266 0.0244 0.0224 0.0206 0.0189 0.0174

2 0.1165 0.1088 0.1016 0.0948 0.0884 0.0824 0.0768 0.0715 0.0666 0.0620

3 0.2513 0.2381 0.2254 0.2133 0.2017 0.1906 0.1800 0.1700 0.1604 0.1512

4 0.4231 0.4061 0.3895 0.3733 0.3575 0.3422 0.3272 0.3127 0.2987 0.2851

5 0.5984 0.5809 0.5635 0.5461 0.5289 0.5119 0.4950 0.4783 0.4619 0.4457

6 0.7474 0.7324 0.7171 0.7017 0.6860 0.6703 0.6544 0.6384 0.6224 0.6063

λ

Chapter 5: Special DistributionPoisson Distribution Table

33

x 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 6.0

7 0.8560 0.8449 0.8335 0.8217 0.8095 0.7970 0.7841 0.7710 0.7576 0.7440

8 0.9252 0.9181 0.9106 0.9027 0.8944 0.8857 0.8766 0.8672 0.8574 0.8472

9 0.9644 0.9603 0.9559 0.9512 0.9462 0.9409 0.9352 0.9292 0.9228 0.9161

10 0.9844 0.9823 0.9800 0.9775 0.9747 0.9718 0.9686 0.9651 0.9614 0.9574

11 0.9937 0.9927 0.9916 0.9904 0.9890 0.9875 0.9859 0.9841 0.9821 0.9799

12 0.9976 0.9972 0.9967 0.9962 0.9955 0.9949 0.9941 0.9932 0.9922 0.9912

13 0.9992 0.9990 0.9988 0.9986 0.9983 0.9980 0.9977 0.9973 0.9969 0.9964

14 0.9997 0.9997 0.9996 0.9995 0.9994 0.9993 0.9991 0.9990 0.9988 0.9986

15 0.9999 0.9999 0.9999 0.9998 0.9998 0.9998 0.9997 0.9996 0.9996 0.9995

16 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998

17 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999

18 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

λ

x 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 7.0

0 0.0022 0.0020 0.0018 0.0017 0.0015 0.0014 0.0012 0.0011 0.0010 0.0009

1 0.0159 0.0146 0.0134 0.0123 0.0113 0.0103 0.0095 0.0087 0.0080 0.0073

2 0.0577 0.0536 0.0498 0.0463 0.0430 0.0400 0.0371 0.0344 0.0320 0.0296

3 0.1425 0.1342 0.1264 0.1189 0.1118 0.1052 0.0988 0.0928 0.0871 0.0818

4 0.2719 0.2592 0.2469 0.2351 0.2237 0.2127 0.2022 0.1920 0.1823 0.1730

5 0.4298 0.4141 0.3988 0.3837 0.3690 0.3547 0.3406 0.3270 0.3137 0.3007

6 0.5902 0.5742 0.5582 0.5423 0.5265 0.5108 0.4953 0.4799 0.4647 0.4497

7 0.7301 0.7160 0.7017 0.6873 0.6728 0.6581 0.6433 0.6285 0.6136 0.5987

8 0.8367 0.8259 0.8148 0.8033 0.7916 0.7796 0.7673 0.7548 0.7420 0.7291

9 0.9090 0.9016 0.8939 0.8858 0.8774 0.8686 0.8596 0.8502 0.8405 0.8305

10 0.9531 0.9486 0.9437 0.9386 0.9332 0.9274 0.9214 0.9151 0.9084 0.9015

11 0.9776 0.9750 0.9723 0.9693 0.9661 0.9627 0.9591 0.9552 0.9510 0.9467

12 0.9900 0.9887 0.9873 0.9857 0.9840 0.9821 0.9801 0.9779 0.9755 0.9730

13 0.9958 0.9952 0.9945 0.9937 0.9929 0.9920 0.9909 0.9898 0.9885 0.9872

14 0.9984 0.9981 0.9978 0.9974 0.9970 0.9966 0.9961 0.9956 0.9950 0.9943

15 0.9994 0.9993 0.9992 0.9990 0.9988 0.9986 0.9984 0.9982 0.9979 0.9976

16 0.9998 0.9997 0.9997 0.9996 0.9996 0.9995 0.9994 0.9993 0.9992 0.9990

17 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998 0.9998 0.9997 0.9997 0.9996

18 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999

19 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

λ

x 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.0

0 0.0008 0.0007 0.0007 0.0006 0.0006 0.0005 0.0005 0.0004 0.0004 0.0003

1 0.0067 0.0061 0.0056 0.0051 0.0047 0.0043 0.0039 0.0036 0.0033 0.0030

2 0.0275 0.0255 0.0236 0.0219 0.0203 0.0188 0.0174 0.0161 0.0149 0.0138

3 0.0767 0.0719 0.0674 0.0632 0.0591 0.0554 0.0518 0.0485 0.0453 0.0424

4 0.1641 0.1555 0.1473 0.1395 0.1321 0.1249 0.1181 0.1117 0.1055 0.0996

5 0.2881 0.2759 0.2640 0.2526 0.2414 0.2307 0.2203 0.2103 0.2006 0.1912

6 0.4349 0.4204 0.4060 0.3920 0.3782 0.3646 0.3514 0.3384 0.3257 0.3134Continue;


34

λ

x 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 8.0

7 0.5838 0.5689 0.5541 0.5393 0.5246 0.5100 0.4956 0.4812 0.4670 0.4530

8 0.7160 0.7027 0.6892 0.6757 0.6620 0.6482 0.6343 0.6204 0.6065 0.5925

9 0.8202 0.8096 0.7988 0.7877 0.7764 0.7649 0.7531 0.7411 0.7290 0.7166

10 0.8942 0.8867 0.8788 0.8707 0.8622 0.8535 0.8445 0.8352 0.8257 0.8159

11 0.9420 0.9371 0.9319 0.9265 0.9208 0.9148 0.9085 0.9020 0.8952 0.8881

12 0.9703 0.9673 0.9642 0.9609 0.9573 0.9536 0.9496 0.9454 0.9409 0.9362

13 0.9857 0.9841 0.9824 0.9805 0.9784 0.9762 0.9739 0.9714 0.9687 0.9658

14 0.9935 0.9927 0.9918 0.9908 0.9897 0.9886 0.9873 0.9859 0.9844 0.9827

15 0.9972 0.9969 0.9964 0.9959 0.9954 0.9948 0.9941 0.9934 0.9926 0.9918

16 0.9989 0.9987 0.9985 0.9983 0.9980 0.9978 0.9974 0.9971 0.9967 0.9963

17 0.9996 0.9995 0.9994 0.9993 0.9992 0.9991 0.9989 0.9988 0.9986 0.9984

18 0.9998 0.9998 0.9998 0.9997 0.9997 0.9996 0.9996 0.9995 0.9994 0.9993

19 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998 0.9998 0.9997

20 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999

21 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

λ

x 8.1 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 9.0

0 0.0003 0.0003 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0001 0.0001

1 0.0028 0.0025 0.0023 0.0021 0.0019 0.0018 0.0016 0.0015 0.0014 0.0012

2 0.0127 0.0118 0.0109 0.0100 0.0093 0.0086 0.0079 0.0073 0.0068 0.0062

3 0.0396 0.0370 0.0346 0.0323 0.0301 0.0281 0.0262 0.0244 0.0228 0.0212

4 0.0940 0.0887 0.0837 0.0789 0.0744 0.0701 0.0660 0.0621 0.0584 0.0550

5 0.1822 0.1736 0.1653 0.1573 0.1496 0.1422 0.1352 0.1284 0.1219 0.1157

6 0.3013 0.2896 0.2781 0.2670 0.2562 0.2457 0.2355 0.2256 0.2160 0.2068

7 0.4391 0.4254 0.4119 0.3987 0.3856 0.3728 0.3602 0.3478 0.3357 0.3239

8 0.5786 0.5647 0.5507 0.5369 0.5231 0.5094 0.4958 0.4823 0.4689 0.4557

9 0.7041 0.6915 0.6788 0.6659 0.6530 0.6400 0.6269 0.6137 0.6006 0.5874

10 0.8058 0.7955 0.7850 0.7743 0.7634 0.7522 0.7409 0.7294 0.7178 0.7060

11 0.8807 0.8731 0.8652 0.8571 0.8487 0.8400 0.8311 0.8220 0.8126 0.8030

12 0.9313 0.9261 0.9207 0.9150 0.9091 0.9029 0.8965 0.8898 0.8829 0.8758

13 0.9628 0.9595 0.9561 0.9524 0.9486 0.9445 0.9403 0.9358 0.9311 0.9261

14 0.9810 0.9791 0.9771 0.9749 0.9726 0.9701 0.9675 0.9647 0.9617 0.9585

15 0.9908 0.9898 0.9887 0.9875 0.9862 0.9848 0.9832 0.9816 0.9798 0.9780

16 0.9958 0.9953 0.9947 0.9941 0.9934 0.9926 0.9918 0.9909 0.9899 0.9889

17 0.9982 0.9979 0.9977 0.9973 0.9970 0.9966 0.9962 0.9957 0.9952 0.9947

18 0.9992 0.9991 0.9990 0.9989 0.9987 0.9985 0.9983 0.9981 0.9978 0.9976

19 0.9997 0.9997 0.9996 0.9995 0.9995 0.9994 0.9993 0.9992 0.9991 0.9989

20 0.9999 0.9999 0.9998 0.9998 0.9998 0.9998 0.9997 0.9997 0.9996 0.9996

21 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998

22 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999

23 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

λ


35

x 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 10.0

0 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0000

1 0.0011 0.0010 0.0009 0.0009 0.0008 0.0007 0.0007 0.0006 0.0005 0.0005

2 0.0058 0.0053 0.0049 0.0045 0.0042 0.0038 0.0035 0.0033 0.0030 0.0028

3 0.0198 0.0184 0.0172 0.0160 0.0149 0.0138 0.0129 0.0120 0.0111 0.0103

4 0.0517 0.0486 0.0456 0.0429 0.0403 0.0378 0.0355 0.0333 0.0312 0.0293

5 0.1098 0.1041 0.0986 0.0935 0.0885 0.0838 0.0793 0.0750 0.0710 0.0671

6 0.1978 0.1892 0.1808 0.1727 0.1649 0.1574 0.1502 0.1433 0.1366 0.1301

7 0.3123 0.3010 0.2900 0.2792 0.2687 0.2584 0.2485 0.2388 0.2294 0.2202

8 0.4426 0.4296 0.4168 0.4042 0.3918 0.3796 0.3676 0.3558 0.3442 0.3328

9 0.5742 0.5611 0.5479 0.5349 0.5218 0.5089 0.4960 0.4832 0.4705 0.4579

10 0.6941 0.6820 0.6699 0.6576 0.6453 0.6329 0.6205 0.6080 0.5955 0.5830

11 0.7932 0.7832 0.7730 0.7626 0.7520 0.7412 0.7303 0.7193 0.7081 0.6968

12 0.8684 0.8607 0.8529 0.8448 0.8364 0.8279 0.8191 0.8101 0.8009 0.7916

13 0.9210 0.9156 0.9100 0.9042 0.8981 0.8919 0.8853 0.8786 0.8716 0.8645

14 0.9552 0.9517 0.9480 0.9441 0.9400 0.9357 0.9312 0.9265 0.9216 0.9165

15 0.9760 0.9738 0.9715 0.9691 0.9665 0.9638 0.9609 0.9579 0.9546 0.9513

16 0.9878 0.9865 0.9852 0.9838 0.9823 0.9806 0.9789 0.9770 0.9751 0.9730

17 0.9941 0.9934 0.9927 0.9919 0.9911 0.9902 0.9892 0.9881 0.9870 0.9857

18 0.9973 0.9969 0.9966 0.9962 0.9957 0.9952 0.9947 0.9941 0.9935 0.9928

19 0.9988 0.9986 0.9985 0.9983 0.9980 0.9978 0.9975 0.9972 0.9969 0.9965

20 0.9995 0.9994 0.9993 0.9992 0.9991 0.9990 0.9989 0.9987 0.9986 0.9984

21 0.9998 0.9998 0.9997 0.9997 0.9996 0.9996 0.9995 0.9995 0.9994 0.9993

22 0.9999 0.9999 0.9999 0.9999 0.9999 0.9998 0.9998 0.9998 0.9997 0.9997

23 1.0000 1.0000 1.0000 1.0000 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999

24 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

λ

x 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0

0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

1 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

2 0.0012 0.0005 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

3 0.0049 0.0023 0.0011 0.0005 0.0002 0.0001 0.0000 0.0000 0.0000 0.0000

4 0.0151 0.0076 0.0037 0.0018 0.0009 0.0004 0.0002 0.0001 0.0000 0.0000

5 0.0375 0.0203 0.0107 0.0055 0.0028 0.0014 0.0007 0.0003 0.0002 0.0001

6 0.0786 0.0458 0.0259 0.0142 0.0076 0.0040 0.0021 0.0010 0.0005 0.0003

7 0.1432 0.0895 0.0540 0.0316 0.0180 0.0100 0.0054 0.0029 0.0015 0.0008

8 0.2320 0.1550 0.0998 0.0621 0.0374 0.0220 0.0126 0.0071 0.0039 0.0021

9 0.3405 0.2424 0.1658 0.1094 0.0699 0.0433 0.0261 0.0154 0.0089 0.0050

10 0.4599 0.3472 0.2517 0.1757 0.1185 0.0774 0.0491 0.0304 0.0183 0.0108

11 0.5793 0.4616 0.3532 0.2600 0.1848 0.1270 0.0847 0.0549 0.0347 0.0214

12 0.6887 0.5760 0.4631 0.3585 0.2676 0.1931 0.1350 0.0917 0.0606 0.0390

13 0.7813 0.6815 0.5730 0.4644 0.3632 0.2745 0.2009 0.1426 0.0984 0.0661

14 0.8540 0.7720 0.6751 0.5704 0.4657 0.3675 0.2808 0.2081 0.1497 0.1049

15 0.9074 0.8444 0.7636 0.6694 0.5681 0.4667 0.3715 0.2867 0.2148 0.1565


36

Continue; λ

x 11.0 12.0 13.0 14.0 15.0 16.0 17.0 18.0 19.0 20.0

16 0.9441 0.8987 0.8355 0.7559 0.6641 0.5660 0.4677 0.3751 0.2920 0.2211

17 0.9678 0.9370 0.8905 0.8272 0.7489 0.6593 0.5640 0.4686 0.3784 0.2970

18 0.9823 0.9626 0.9302 0.8826 0.8195 0.7423 0.6550 0.5622 0.4695 0.3814

19 0.9907 0.9787 0.9573 0.9235 0.8752 0.8122 0.7363 0.6509 0.5606 0.4703

20 0.9953 0.9884 0.9750 0.9521 0.9170 0.8682 0.8055 0.7307 0.6472 0.5591

21 0.9977 0.9939 0.9859 0.9712 0.9469 0.9108 0.8615 0.7991 0.7255 0.6437

22 0.9990 0.9970 0.9924 0.9833 0.9673 0.9418 0.9047 0.8551 0.7931 0.7206

23 0.9995 0.9985 0.9960 0.9907 0.9805 0.9633 0.9367 0.8989 0.8490 0.7875

24 0.9998 0.9993 0.9980 0.9950 0.9888 0.9777 0.9594 0.9317 0.8933 0.8432

25 0.9999 0.9997 0.9990 0.9974 0.9938 0.9869 0.9748 0.9554 0.9269 0.8878

26 1.0000 0.9999 0.9995 0.9987 0.9967 0.9925 0.9848 0.9718 0.9514 0.9221

27 1.0000 0.9999 0.9998 0.9994 0.9983 0.9959 0.9912 0.9827 0.9687 0.9475

28 1.0000 1.0000 0.9999 0.9997 0.9991 0.9978 0.9950 0.9897 0.9805 0.9657

29 1.0000 1.0000 1.0000 0.9999 0.9996 0.9989 0.9973 0.9941 0.9882 0.9782

30 1.0000 1.0000 1.0000 0.9999 0.9998 0.9994 0.9986 0.9967 0.9930 0.9865

31 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9993 0.9982 0.9960 0.9919

32 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9996 0.9990 0.9978 0.9953

33 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9995 0.9988 0.9973

34 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9994 0.9985

35 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9997 0.9992

36 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998 0.9996

37 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999 0.9998

38 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999

39 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9999

40 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000


37

STANDARD NORMAL DISTRIBUTION2

21

( )2

z x

P Z z e dxπ

−

−∞

≤ = ∫

z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

0.0 0.5000 0.5040 0.5080 0.5120 0.5160 0.5199 0.5239 0.5279 0.5319 0.5359

0.1 0.5398 0.5438 0.5478 0.5517 0.5557 0.5596 0.5636 0.5675 0.5714 0.5753

0.2 0.5793 0.5832 0.5871 0.5910 0.5948 0.5987 0.6026 0.6064 0.6103 0.6141

0.3 0.6179 0.6217 0.6255 0.6293 0.6331 0.6368 0.6406 0.6443 0.6480 0.6517

0.4 0.6554 0.6591 0.6628 0.6664 0.6700 0.6736 0.6772 0.6808 0.6844 0.6879

0.5 0.6915 0.6950 0.6985 0.7019 0.7054 0.7088 0.7123 0.7157 0.7190 0.7224

0.6 0.7257 0.7291 0.7324 0.7357 0.7389 0.7422 0.7454 0.7486 0.7517 0.7549

0.7 0.7580 0.7611 0.7642 0.7673 0.7704 0.7734 0.7764 0.7794 0.7823 0.7852

0.8 0.7881 0.7910 0.7939 0.7967 0.7995 0.8023 0.8051 0.8078 0.8106 0.8133

0.9 0.8159 0.8186 0.8212 0.8238 0.8264 0.8289 0.8315 0.8340 0.8365 0.8389

1.0 0.8413 0.8438 0.8461 0.8485 0.8508 0.8531 0.8554 0.8577 0.8599 0.8621

1.1 0.8643 0.8665 0.8686 0.8708 0.8729 0.8749 0.8770 0.8790 0.8810 0.8830

1.2 0.8849 0.8869 0.8888 0.8907 0.8925 0.8944 0.8962 0.8980 0.8997 0.9015

1.3 0.9032 0.9049 0.9066 0.9082 0.9099 0.9115 0.9131 0.9147 0.9162 0.9177

1.4 0.9192 0.9207 0.9222 0.9236 0.9251 0.9265 0.9279 0.9292 0.9306 0.9319

1.5 0.9332 0.9345 0.9357 0.9370 0.9382 0.9394 0.9406 0.9418 0.9429 0.9441

1.6 0.9452 0.9463 0.9474 0.9484 0.9495 0.9505 0.9515 0.9525 0.9535 0.9545

1.7 0.9554 0.9564 0.9573 0.9582 0.9591 0.9599 0.9608 0.9616 0.9625 0.9633

1.8 0.9641 0.9649 0.9656 0.9664 0.9671 0.9678 0.9686 0.9693 0.9699 0.9706

1.9 0.9713 0.9719 0.9726 0.9732 0.9738 0.9744 0.9750 0.9756 0.9761 0.9767

2.0 0.9772 0.9778 0.9783 0.9788 0.9793 0.9798 0.9803 0.9808 0.9812 0.9817

2.1 0.9821 0.9826 0.9830 0.9834 0.9838 0.9842 0.9846 0.9850 0.9854 0.9857

2.2 0.9861 0.9864 0.9868 0.9871 0.9875 0.9878 0.9881 0.9884 0.9887 0.9890

2.3 0.9893 0.9896 0.9898 0.9901 0.9904 0.9906 0.9909 0.9911 0.9913 0.9916

2.4 0.9918 0.9920 0.9922 0.9925 0.9927 0.9929 0.9931 0.9932 0.9934 0.9936

2.5 0.9938 0.9940 0.9941 0.9943 0.9945 0.9946 0.9948 0.9949 0.9951 0.9952

2.6 0.9953 0.9955 0.9956 0.9957 0.9959 0.9960 0.9961 0.9962 0.9963 0.9964

2.7 0.9965 0.9966 0.9967 0.9968 0.9969 0.9970 0.9971 0.9972 0.9973 0.9974

2.8 0.9974 0.9975 0.9976 0.9977 0.9977 0.9978 0.9979 0.9979 0.9980 0.9981

2.9 0.9981 0.9982 0.9982 0.9983 0.9984 0.9984 0.9985 0.9985 0.9986 0.9986

3.0 0.9987 0.9987 0.9987 0.9988 0.9988 0.9989 0.9989 0.9989 0.9990 0.9990

3.1 0.9990 0.9991 0.9991 0.9991 0.9992 0.9992 0.9992 0.9992 0.9993 0.9993

3.2 0.9993 0.9993 0.9994 0.9994 0.9994 0.9994 0.9994 0.9995 0.9995 0.9995

3.3 0.9995 0.9995 0.9995 0.9996 0.9996 0.9996 0.9996 0.9996 0.9996 0.9997

3.4 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9997 0.9998

3.5 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998 0.9998

3.6 0.9998 0.9998 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999

3.7 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999

3.8 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999 0.9999

3.9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

Chapter 5: Special DistributionStandard Normal Distribution Table

0 z

38

Continue;z 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09

-0.0 0.5000 0.4960 0.4920 0.4880 0.4840 0.4801 0.4761 0.4721 0.4681 0.4641

-0.1 0.4602 0.4562 0.4522 0.4483 0.4443 0.4404 0.4364 0.4325 0.4286 0.4247

-0.2 0.4207 0.4168 0.4129 0.4090 0.4052 0.4013 0.3974 0.3936 0.3897 0.3859

-0.3 0.3821 0.3783 0.3745 0.3707 0.3669 0.3632 0.3594 0.3557 0.3520 0.3483

-0.4 0.3446 0.3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 0.3156 0.3121

-0.5 0.3085 0.3050 0.3015 0.2981 0.2946 0.2912 0.2877 0.2843 0.2810 0.2776

-0.6 0.2743 0.2709 0.2676 0.2643 0.2611 0.2578 0.2546 0.2514 0.2483 0.2451

-0.7 0.2420 0.2389 0.2358 0.2327 0.2296 0.2266 0.2236 0.2206 0.2177 0.2148

-0.8 0.2119 0.2090 0.2061 0.2033 0.2005 0.1977 0.1949 0.1922 0.1894 0.1867

-0.9 0.1841 0.1814 0.1788 0.1762 0.1736 0.1711 0.1685 0.1660 0.1635 0.1611

-1.0 0.1587 0.1562 0.1539 0.1515 0.1492 0.1469 0.1446 0.1423 0.1401 0.1379

-1.1 0.1357 0.1335 0.1314 0.1292 0.1271 0.1251 0.1230 0.1210 0.1190 0.1170

-1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985

-1.3 0.0968 0.0951 0.0934 0.0918 0.0901 0.0885 0.0869 0.0853 0.0838 0.0823

-1.4 0.0808 0.0793 0.0778 0.0764 0.0749 0.0735 0.0721 0.0708 0.0694 0.0681

-1.5 0.0668 0.0655 0.0643 0.0630 0.0618 0.0606 0.0594 0.0582 0.0571 0.0559

-1.6 0.0548 0.0537 0.0526 0.0516 0.0505 0.0495 0.0485 0.0475 0.0465 0.0455

-1.7 0.0446 0.0436 0.0427 0.0418 0.0409 0.0401 0.0392 0.0384 0.0375 0.0367

-1.8 0.0359 0.0351 0.0344 0.0336 0.0329 0.0322 0.0314 0.0307 0.0301 0.0294

-1.9 0.0287 0.0281 0.0274 0.0268 0.0262 0.0256 0.0250 0.0244 0.0239 0.0233

-2.0 0.0228 0.0222 0.0217 0.0212 0.0207 0.0202 0.0197 0.0192 0.0188 0.0183

-2.1 0.0179 0.0174 0.0170 0.0166 0.0162 0.0158 0.0154 0.0150 0.0146 0.0143

-2.2 0.0139 0.0136 0.0132 0.0129 0.0125 0.0122 0.0119 0.0116 0.0113 0.0110

-2.3 0.0107 0.0104 0.0102 0.0099 0.0096 0.0094 0.0091 0.0089 0.0087 0.0084

-2.4 0.0082 0.0080 0.0078 0.0075 0.0073 0.0071 0.0069 0.0068 0.0066 0.0064

-2.5 0.0062 0.0060 0.0059 0.0057 0.0055 0.0054 0.0052 0.0051 0.0049 0.0048

-2.6 0.0047 0.0045 0.0044 0.0043 0.0041 0.0040 0.0039 0.0038 0.0037 0.0036

-2.7 0.0035 0.0034 0.0033 0.0032 0.0031 0.0030 0.0029 0.0028 0.0027 0.0026

-2.8 0.0026 0.0025 0.0024 0.0023 0.0023 0.0022 0.0021 0.0021 0.0020 0.0019

-2.9 0.0019 0.0018 0.0018 0.0017 0.0016 0.0016 0.0015 0.0015 0.0014 0.0014

-3.0 0.0013 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011 0.0011 0.0010 0.0010

-3.1 0.0010 0.0009 0.0009 0.0009 0.0008 0.0008 0.0008 0.0008 0.0007 0.0007

-3.2 0.0007 0.0007 0.0006 0.0006 0.0006 0.0006 0.0006 0.0005 0.0005 0.0005

-3.3 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0004 0.0004 0.0003

-3.4 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0003 0.0002

-3.5 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002 0.0002

-3.6 0.0002 0.0002 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001

-3.7 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001

-3.8 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001

-3.9 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000


39

ANSWERS

EXERCISE 1

1. p = 0.05, q = 1 – 0.05 = 0.95, n = 6

a) P(X=3) = 6C3(0.05)3(0.95)6-3

= 0.0021

b) P(X < 2) = P(X = 0) + P(X = 1) = 6C0(0.05)0(0.95)6-0 + 6C1(0.05)1(0.95)6-1

= 0.7351 +0.2321 = 0.9672

c) P(X=0) = 6C0(0.05)0(0.95)6-0

= 0.7351

2. p = 0.30, q = 1 – 0.30 = 0.70, n = 5

P(X ≥ 3) = P(X = 3) + P(X = 4) + P(X=5)

= 5C3 (0.30)3(0.70)2 + 5C4 (0.30)4(0.70)1 +

5C5 (0.30)5(0.70)0

= 0.1323 + 0.0283 + 0.0024

= 0.1631

3. p = 0.40, q = 1 – 0.40 = 0.60, n = 5

a) P(X=2) = 5C2(0.04)2(0.6)5-2

= 0.3456

b) P(X ≤ 3) = 1- ( P(X = 4) + P(X = 5))= 1 – (5C4 (0.4)4(0.6)1 +

5C5 (0.4)5(0.6)0)= 1 – (0.0768 +0.01024= 0.91296

c) P( X ≥ 2 ) = P(X = 0) + P(X=1) + ( P=2)= 5C0 (0.4)0(0.6)5 +

5C1 (0.4)1(0.6)4 + 5C2 (0.4)2(0.6)3

= 0.07776 + 0.2592 + 0.3456= 0.6826

4. p = 0.6, q = 1 – 0.6 = 0.4, n = 10

P(X=3) = 10C3 (0.6)3(0.4)7

= 0.042

5. p = 0.65, q = 0.35, n = 10

a) P(X = 4) = 10C4 (0.65)4(0.35)10-4

= 0.0689

p = 0.35, q = 0.65, n = 10

b) P(X ≥ 6) = 1 – P(X ≤ 5) = 1 – 0.9051 = 0.0949

6. p = 0.4, q = 0.6, n = 15

a) P(X ≤ 5) = 0.4032

b) P(X < 4) = P(X ≤ 3) = 0.0905

c) P(X ≥ 9) = 1 – P(X ≤ 8) = 1 – 0.9050 = 0.095

d) P(6 ≤ X ≤ 9) = P(X ≤ 9) – P(X ≤ 5) = 0.9662 – 0.4032

= 0.5630

e) P(4 < X < 8) = P( X ≤ 7) – P(X ≤ 4) = 0.7869 – 0.2173 = 0.5696

EXERCISE 2

1. µ = np = 200(0.83) = 166

σ2 = npq = 200(0.83)(0.17) = 28.22

σ = √npq = √28.22 = 5.3122

2. µ = np = 1000(0.25) = 250

σ2 = npq = 1000(0.25)(0.75) = 187.5

σ = √npq = √187.5 = 13.6931


40

EXERCISE 3

1. λ = 2 telemarketing phone calls per week

a) P(X = 6) = 2 62

6!

e−

= 0.0120

λ = 8 telemarketing phone calls in one month

b) P(X < 3) = P(X ≤ 2)

= P(X = 0) + P(X = 1) +

P(X = 2)

= 8 0 8 1 8 28 8 8

0! 1! 2!

e e e− − −

+ +

= 0.0003 + 0.0027 + 0.0107

= 0.0137

2. λ = 3 times breakdown per month

a) 3 23

P(X=2)=2!

e−

= 0.2240

b) P(X ≤ 1) = 3 0 3 13 3

0! 1!

e e− −

+

= 0.0498 + 0.1494

= 0.1992

λ = 6 times breakdown in 2 months

c) P(X = 0) = 6 06

0!

e−

= 0.0025

3. λ = 0.9 plain lands at the airport per 5 min interval.

a) P(X ≤ 1) = P(X=0) + P(X=1)

= (e-0.9 (0.9)0)/0! + (e-0.9 (0.9)1)/1!

= 0.4066 + 0.3659

= 0.772

λ = 0.9*3 = 2.7 plains land at the airport per 15 min

interval.

b) P(X > 3) = 1- P (X ≤ 3) = 1 – ((P(X=0) + P(X=1) + P(X=2) +

P(X=3) = 1 – ((e-2.7 (2.7)0)/0! +

(e-2.7 (2.7)1)/1!)/1! + (e-2.7 (2.7)2)/2!) + (e-2.7 (2.7)3)/3!)

= 1 – (0.0672 + 0.1815 + 0.245 + 0.2205) = 1 – 0.7142 = 0.286

4. λ = 4.8 customer every half hour

a) P(X = 2) = P(X ≤ 2) – P(X ≤ 1) = 0.1425 – 0.0477 = 0.0948

b) P(X ≤ 2) = 0.1425

c) P(X = 0) = P(X ≤ 0) = 0.0082

λ = 9.6 customer in one hour

d) P(X > 5) = 1 – P(X ≤ 5) = 1 – 0.0838 = 0.9162 5. λ = 8 calls per hour

a) P (X ≥ 8) = 1 – P(X ≤ 7) = 1 – 0.4530 = 0.547

b) P (X ≥ 3) = 1 - P(X ≤ 3) = 1 – 0.0138 = 0.9862

c) P (X ≤ 5) = 0.1912


41


42

EXERCISE 4

a) To the left of z = -2.87P(z < -2.87) = 0.0021

b) To the right of z = 2.45P(z > 2.45) = 1 - P(z < 2.45) = 1 - 0.9929

= 0.0071

c) Between z = -2.15 and z = 1.67P(-2.15 < z < 1.67)= P(z < 1.67) – P(z < -2.15)= 0.9525 – 0.0158 = 0.9367

EXERCISE 5

1. a) P(X < 150) = P(z < 150 - 190) 21

= P(z < -1.9)

= 0.0281

b) P(205 < X < 245)

= P(z < 2.62) - P(z < 0.71)

= 0.9956 – 0.7611

= 0.2345

2. P(X < 3.5) = P(z < 3.5 – 3.1) 0.5

= P(z < 0.8)

= 1 - 0.7881

= 0.2119

= 21.19%

3. n = 6000, μ = 55, σ = 10

a) P(X > 75) = P(z < 75 – 55) 10

= P(z < 2)

= 1 - P(z < 2)

= 1 - 0.9772

= 0.0228

Thus, the number of graduates with distinction

is

0.0228( 6000) = 136.8

≈ 137 graduates

b) P(65 < X < 45) = P[(65 – 55) <z < 45 – 55)] 10 10

= P( 1 < z < -1)

= 0.8413 – 0.1587

= 0.6826

-2.87 0

0 2.45

-2.15 1.67

EXERCISE 6

1. p = 0.1, q = 0.9, n = 5

a) P(X=2) = 5C2(0.1)2(0.9)3

= 0.0729or

P(X=2) = P(X≤2) – P(X≤1) = 0.9914 – 0.9188 = 0.0729

b) P (X ≥ 1) = 1 – P(X=0) = 1 – [5C0(0.1)0(0.9)5 ] = 1 – 0.5905 = 0.4095

or P(X≥ 1) = 1 – P(X≤0)

= 1 – 0.5905 = 0.4095

2. p = 0.2, q = 0.8, n = 20

a) P(X < 3) = P(X≤2)= 0.2061or

P(X < 3) = P(X=0) + P(X=1) + P(X=2)= 20C0(0.2)0(0.8)20 + 20C1(0.2)1(0.8)19 +

20C2(0.2)2(0.8)18

= 0.01152 + 0.0576 + 0.1369= 0.2061

b) P(X=4) = 20C4(0.2)4(0.8)16 = 0.2182or

P(X=4) = P(X≤4) - P(X≤3)= 6296 – 0.4114= 0.2061

3. p = 0.4, q = 0.6, n =20

P(X>10) = 1 – P(X≤9) = 1 – 0.4044

= 0.5956

4. p = 0.12, q = 0.88, n = 75

µ = np = 75(0.12) = 9

σ2 = npq = 75(0.12)(0.88) = 7.92

σ = √npq = √7.92 = 2.814

5. p = 0.88, q = 0.12, n = 16

a) P(X=9) = 16C9(0.88)9(0.12)7

= 0.0013

p = 0.08, q = 0.92, n = 16

b) P(X=4) = 16C4(0.08)4(0.92)12

= 0.0274

p = 0.04, q = 0.96, n = 16

c) P(X=3) = 16C3(0.04)3(0.96)13

= 0.0211

6. λ = 3 loan applications per week

a) P(X = 0) = (e-3 (3)0)/0! = 0.0498

b) P(2 ≤ z ≤ 5) = P(X≤5) - P(X≤2) = 0.9161 – 0.4232 = 0.4929

λ = 6 loan applications per 2 weeks

c) P (X ≥ 5) = 1 - P(X ≤ 4) = 1 – 0.2851 = 0.7149

7. λ = 5 books from each lot

P(X = 5) = (e-5 (5)5)/5! = 0.1754

8. λ = 42/60 = 0.7 customer per minute

a) P(X = 0) = (e-0.7 (0.7)0)/0! = 0.4966

λ = 0.7(5) = 3.5 customer per five minutes

b) P (X ≥ 4) = 1 - P(X ≤ 3) = 1 – 0.5366 = 0.4634

λ = 0.7(10) = 3.5 customer per ten minutes

c) P (2 < X < 6) = P(X ≤ 5) - P(X ≤ 2) = 0.8576 – 0.3208 = 0.5368

9. μ = 2.8, σ = 4

a) P(2 < X < 3) =P(X<3) - P(X<2) =P[((3 – 2.8)/4) <z < ((2 – 2.8)/4]

= P(0.05 < z < -0.2)= 0.8413 – 0.4207= 0.4206

b) P(X<2) = P[z < (2 - 2.8)/4] = P (z < -2) = 0.0228

c) P(X >3) = 1 - P(X<3) = 1 – P [z < (3 - 2.8)/4] = 1 – P (z < 0.05) = 1 – 0.5199 = 0.4801

d) P(X >3.7) = 1 - P(X<3.7) = 1 – P [z < (3.7 – 2.8)/4]

= 1 – P (z < 0.23) = 1 – 0.5910 = 0.4090

Thus, the no. of 1st class honors student is:0.4090(1000) = 409 1st class honors students

10. μ = 24, σ = 3.8

a) P(X >30) = 1 - P(X<30) = 1 – P [z < (30 – 24)/3.8]

= 1 – P (z < 1.58) = 1 – 0.9394 = 0.0606

b) P(20 < X < 25) =P(X≤25) - P(X≤20)=P[((25 – 24)/3.8) <z < ((20 – 24)/3.8]

= P( 0.26 < z < -1.05) = 0.6026 – 0.1469 = 0.4557

c) P(X >25) = 1 - P(X<25) = 1 – P [z < (25 – 24)/3.8]

= 1 – P (z < 0.26) = 1 – 0.6026 = 0.3974

d) P(X<10) = P[z < (2 - 2.8)/04] = P (z < -2) = 0.0228

11. μ = 15, σ = 3

a) P(X<13) = P[z < (13 - 15)/3] = P (z < -0.67) = 0.2514

b) P(X >16) = 1 - P(X<16) = 1 – P [z < (16 – 15)/3]

= 1 – P (z < 0.33) = 1 – 0.6293

= 0.3707

c) P(14 < X < 17) =P(X<17) - P(X<14) =P[((17 – 15)/3) <z < ((14 – 15)/3]

= P(0.67 < z < -0.33)= 0.7486 – 0.3707= 0.3779

12. μ = 132, σ = 8

a) P(X >130) = 1 - P(X<130) = 1 – P [z < (130 – 132)/8]

= 1 – P (z < -0.25) = 1 – 0.4013 = 0.5987

b) P(X<140) = P[z < (140 - 132)/8] = P (z < 1.00) = 0.8413

c) P(131 < X < 136) =P(X<136) - P(X<131)=P[((136 – 132)/8) <z < ((131 – 132)/8]

= P(0.5 < z < -0.13) = 0.6915 – 0.4483 = 0.2432

13. μ = 80, σ = 12

a) P(X<90) = P[z < (90 - 82)/12] = P (z < 0.67) = 0.7486

b) P(X >75) = 1 - P(X<75) = 1 – P [z < (75 – 80)/12]

= 1 – P (z < -0.42) = 1 – 0.3372 = 0.6628

c) P(79 < X < 95) =P(X<95) - P(X<79)=P[(95 – 80)/12) <z < (79 – 80)/12]

= P(1.25 < z < -0.08) = 0.8944 – 0.4681 = 0.4263

Matrix No: ____________________ Group: __________

TUTORIAL CHAPTER 5

QUESTION 1

A past history shows that 60% of college students are smokers. A sample of 5 students is randomly selected. What is the probability that exactly one student is a smoker?

(2 marks)

QUESTION 2

Hope.fm Radio Company receives, on the average, two short messaging services (sms) within five minutes from listener. Find the probability that it will receive the following

i. at least four sms within five minutes (2 marks)

ii. two sms within fifteen minutes (3 marks)

Chapter 5: Special DistributionsTutorial

47

14

QUESTION 3

EJ Utara Transport determined that on an annual basis, the distance travel per lorry is normally distributed, with a mean of 50 thousand kilometers and a standard deviation of 10 thousand kilometers. Find the

i. probability that a randomly selected lorry travels from 36 to 50 thousand kilometers in the year

(3 marks)

ii. minimum distance traveled by 80% of the lorries. (4 marks)

Chapter 5: Special DistributionsTutorial

48

sqqs1013 ch5-a122

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