khatijahhusna binti abd rani eqt271 sem ii 2014/2015 chapter 1 part 2 slide was prepared by miss...

KHATIJAHHUSNA BINTI ABD RANI EQT271 SEM II 2014/2015 CHAPTER 1 PART 2 Slide was prepared by Miss Syafawati (with modification) PROBABILITY DISTRIBUTIONS

DISCRETECONTINUOUS BinomialPoisson Normal

A variable whose values are determined by chance Random Process

DISCRETE CONTINUOUS

A probability function is a function which assigns probabilities to the values of a random variable. Individual probability values may be denoted by the symbol P(X=x), in the discrete case, which indicates that the random variable can have various specific values. All the probabilities must be between 0 and 1; 0 P(X=x) 1. The sum of the probabilities of the outcomes must be 1. P(X=x)=1 It may also be denoted by the symbol f(x), in the continuous, which indicates that a mathematical function is involved. Probability Distributions

X=number of heads after 3 flips a fair coin HHHTHH HHTTHT HTHTTH HTTTTT Probability

Check whether the distribution is a probability distribution. Solution # so the distribution is not a probability distribution. X01234 P(X=x)0.1250.3750.0250.3750.125

An experiment in which satisfied the following characteristic is called a binomial experiment: 1. There must be a fixed number of trials. 2. Each trial can result in one of two outcomes, which we denote by success, S or failure, F. 3. The outcomes of each trial are independent. 4. The probability of success is constant from trial to trial, we denote the probability of success by p and the probability of failure is equal to (1 - p) = q. Examples: 1. No. of getting a head in tossing a coin 10 times. 2. No. of getting a six in tossing 7 dice. 3. A firm bidding for contracts will either get a contract or not

A binomial experiment consist of n identical trial with probability of success, p in each trial. The probability of x success in n trials is given by The Mean and Variance of X if X ~ B(n,p) are Mean : Variance : Std Deviation : where n is the total number of trials, p is the probability of success and q is the probability of failure.

Solutions:

Cumulative Binomial Distribution When the sample is relatively large, tables of Binomial are often used. Since the probabilities provided in the tables are in the cumulative form the following guidelines can be used:

In a Binomial Distribution, n =12 and p = 0.3. Find the following probabilities.

0123456789101112

0123456789101112 0123456789101112

0123456789101112

In August 2009, David and Maria conducted a survey for Fortune magazine to examine CEO`s attitudes toward employee`s personal problems. 30% of the CEOs interviewed felt that personal problems were none of the company`s business. Assume that this result is true for the current population of CEOs. Using the Binomial distribution tables, in a random samples of 15, find the probability that (i) The number of CEOs who hold this view is 10. (ii) The number of CEOs who hold this view is between 9 and 12. (iii) The number of CEOs who hold this view is at most 7. (iv) Find the mean and standard deviation of binomial distribution.

Solution: i) ii) iii) iv)

Poisson distribution is the probability distribution of the number of successes in a given space*. *space can be dimensions, place or time or combination of them Used when n is large and p is small & when the independent variables occur over a period of time Examples: 1. No. of cars passing a toll booth in one hour. 2. No. defects in a square meter of fabric 3. No. of network error experienced in a day.

A random variable X has a Poisson distribution and it is referred to as a Poisson random variable if and only if its probability distribution is given by A random variable X having a Poisson distribution can also be written as

Consider a Poisson random variable with. Calculate the following probabilities : a) Write the distribution of Poisson b) c) d)

Solution: a) b) c) d)

Granma bakes chocolate chip cookies in batches 0f 100. She puts 300 chips into the dough. When the cookies are done, she gives you one. What is the probability that your cookie contains no chocolate chips?

The Poisson distribution is suitable as an approximation of Binomial probabilities when n is large and p is small. Approximation can be made when, and either or

Suppose a life insurance company insures the lives of 4000 men aged 42. If actuarial studies show the probability that any 42 year old man will die in a given year to be 0.001, find the exact probability that the company will have to pay x = 4 claims during a given year.

Solution: Step: 1. Write the distribution of Binomial 2. The value for n is large and value of p is too small, check whether 3. If yes, proceed to solve using Poisson Approximation. Use formula

Read the following questions and decide whether the Poisson or the Binomial distribution should be used to answer it. 1. A typist makes on average 2 mistakes per page. What is the probability of a particular page having no errors on it?. 2. A computer crashes once every 2 days on average. What is the probability of there being 2 crashes in one week? 3. Components are packed in boxes of 20. The probability of a component being defective is 0.1. What is the probability of a box containing 2 defective components?

4. ICs are packaged in boxes of 10. The probability of an ic being faulty is 2%. What is the probability of a box containing 2 faulty ics? 5. The mean number of faults in a new house is 8. What is the probability of buying a new house with exactly 1 fault?

Numerous continuous variables have distribution closely resemble the normal distribution. The normal distribution can be used to approximate various discrete probability distribution.

CHARACTERISTICS OF NORMAL DISTRIBUTION Bell Shaped Symmetrical Mean, Median and Mode are Equal Location is determined by the mean, Spread is determined by the standard deviation, The random variable has an infinite theoretical range: + to Mean = Median = Mode X f(X)

By varying the parameters and , we obtain different normal distributions Many Normal Distributions

The Standard Normal Distribution Any normal distribution (with any mean and standard deviation combination) can be transformed into the standard normal distribution (Z) Need to transform X units into Z units using The standardized normal distribution (Z) has a mean of, and a standard deviation of 1, Z is denoted by Thus, its density function becomes

Calculating Probabilities for a General Normal Random Variable Mostly, the probabilities involved x, a normal random variable with mean, and standard deviation, Then, you have to standardized the interval of interest, writing it in terms of z, the standard normal random variable. Once this is done, the probability of interest is the area that you find using the standard normal probability distribution. Normal probability distribution, Need to transform x to z using

Patterns for Finding Areas under the Standard Normal Curve

Z table

a) Find the area under the standard normal curve of a) Find the area under the standard normal curve of 01

Exercise 1.6 Determine the probability or area for the portions of the Normal distribution described. Answer : a) 0.1736, b) 0.4783, c) 0.8078, d) 0.9812, e) 0.0614

Z table

Exercise 1.7

Z table Solutions:

Suppose X is a normal distribution N(25,25). Find Solutions

Exercise 1.8 1. Suppose X is a normal distribution, N(70,4). Find a) b) 2. Suppose the test scores of 600 students are normally distributed with a mean of 76 and standard deviation of 8. The number of students scoring is from 70 to 82 is: Answer : 1. a) 0.927 b) 0.0228 2. 328 students

Normal Approximation of the Binomial Distribution When the number of observations or trials n in a binomial experiment is relatively large, the normal probability distribution can be used to approximate binomial probabilities. A convenient rule is that such approximation is acceptable when

Continuous Correction Factor The continuous correction factor needs to be made when a continuous curve is being used to approximate discrete probability distributions. 0.5 is added or subtracted as a continuous correction factor according to the form of the probability statement as follows :

How do calculate Binomial Probabilities Using the Normal Approximation? Find the necessary values of n and p. Calculate Write the probability you need in terms of x. Correct the value of x with appropriate continuous correction factor (ccf). Convert the necessary x-values to z-values using Use Standard Normal Table to calculate the approximate probability.

In a certain country, 45% of registered voters are male. If 300 registered voters from that country are selected at random, find the probability that at least 155 are males. Solutions:

Suppose that 5% of the population over 70 years old has disease A. Suppose a random sample of 9600 people over 70 is taken. What is the probability that less than 500 of them have disease A? Answer: 0.8186 Exercise 1.9

Normal Approximation of the Poisson Distribution When the mean of a Poisson distribution is relatively large, the normal probability distribution can be used to approximate Poisson probabilities. A convenient rule is that such approximation is acceptable when

A grocery store has an ATM machine inside. An average of 5 customers per hour comes to use the machine. What is the probability that more than 30 customers come to use the machine between 8.00 am and 5.00 pm? Solution:

Exercise 1.10 The average number of accidental drowning in United States per year is 3.0 per 100000 population. Find the probability that in a city of population 400000 there will be less than 10 accidental drowning per year. Answer : 0.2358

Exercise 1.11 1. Reported that the mean weekly income of a shift foreman in the glass industry is normally distributed with a mean of $1000 and standard deviation of $100. What is the probability of selecting a shift foreman in the glass industry whose income is a) Between $1000 and $1100. b) Between $790 and $1000. c) Between $840 and $1200. Answer : a) 0.3413, b) 0.4821, c) 0.9224

khatijahhusna binti abd rani eqt271 sem ii 2014/2015 chapter 1 part 2 slide was prepared by miss...

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