chapter 4. discrete random variables a random variable is a way of recording a quantitative variable...

Chapter 4. Discrete Random Variables

A random variable is a way of recording a quantitative variable of a random experiment.

A variable which can take on only finitely many different values is called discrete.

Example: The number of girls in a family of 8 children

Example: The number of seeds which successfully germinate when 50 seeds are planted

Continuous random variables

• A random variable which can take on any value (ie, all values) in a certain interval is called a continuous random variable.

• EX. The height in centimeters of a 16 year old Canadian male.

• Ex. The dosage in ml. of a certain pain killer

Example

• Let 3 coins be tossed and let x denote the number of heads

• Possible values for x are 0, 1, 2, and 3,

• As done earlier, it is easy to compute

• Pr(0) = 1/8, Pr(3) = 1/8, and Pr(1) = Pr(2) = 3/8

• Notation: We will also use the notation

• P(x = 0) = 1/8, and so on.

Properties of Probability, P( X = xi )

1)(0 (1) ixXP

1)( (2)1

n

iixXP

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Definition

Example

Graph the probability distribution of the random variable obtained by flipping an unbiased coin two times and counting the number of times heads comes up.

Solution

• Possible values of x are 0, 1, 2, and a quick check shows P(0) = ¼, P(1) = 1/2, and P(2) = ¼.


Probability distribution for a two-coin toss


Definition


Procedure


Figure 4.6 Shapes of two probability distributions for a discrete random variable x

Example

• Medical research shows a certain type of chemotherapy is successful 70% of the time. Suppose 5 patients are treated and let x denote the number of successes. One can show

• x 0 1 2 3 4 5

P(x) .002 .029 .132 .309 .360 .168


Graph of p(x)

Find the mean and interpret

• Applying the formulae we obtain

• Mean = 3.50

• Consider a large number of trials, each consisting of treating 5 patients. On average, the number of successes will be 3.5. Thus, if 200 trials each of 5 patients is conducted, we would expect and average of 3.5 successes per trial for a total of 700 successes for 1000 patients

Find the standard deviation and interpret

• Using the formula you can check that• Standard deviation = 1.02.• Using Empirical Rule, would expect that

approximately 68% of times the trial is repeated, the outcome will be between 3.5-1.02 and 3.5+1.02, i.e., will lie in the interval [2.48, 4.52].

• What is the actual percentage of times the outcome will be in that interval?

Binomial Experiment

A binomial experiment is one that:

1) Has a fixed number of trials (n)

2) These trials are independent

3) Each trial must have all outcomes classified into two categories (Success or Failure)

4) The probability of success remains constant for all trials.

Notation:

• S = success and P(S) = p

• F = Failure and P(F) = q = 1- p

• n = fixed number of trials

• x = specific number of successes in n trials

• P(x) = the probability of getting exactly x successes among n trials

Factorials

0! = 1

1! = 1

2! = 2 * 1

3! = 3 * 2 * 1

4! = 4* 3 * 2 * 1

n! = n*(n-1)!

Factorials

0! = 1

1! = 1

2! = 2 * 1=2

3! = 3 * 2 * 1=6

4! = 4* 3 * 2 * 1=24

n! = n*(n-1)!

Binomial Probability Distribution

In a binomial experiment, with constant probability p of success at each trial, the probability of x successes in n trials is given by

xnxqpxxn

nsuccessesxP

!)!(

!) (

ExampleShaq is a basketball player who takes a lot of free throws. The probability of Shaq making a free throw is 0.60 on each throw.

With 3 free throws what is the probability that he makes 2 shots?

Shaq is a basketball player who takes a lot of free throws. The probability of Shaq making a free throw is 0.60 on each throw.

With 3 free throws what is the probability that he makes 2 shots?

0.432

)4(.)6(.!2)!23(

!3)2( 232

xP

Example

Example

Flipping a biased coin 8 times. The probability of heads on each trial is 0.4. What is the probability of obtaining at least 2 heads.

Example


)8(...)3()2()2( xPxPxPxP

Example


)1(1

)8(...)3()2()2(

xP

xPxPxPxP

Example


)0()1(1

)1(1

)8(...)3()2()2(

xPxP

xP

xPxPxPxP

ExampleFlipping a biased coin 8 times. The probability of heads on each trial is 0.4. What is the probability of obtaining at least 2 heads.

8071 )6(.)4(.!0 !8

!8)6(.)4(.

!1 !7

!81

)0()1(1

)1(1

)8(...)3()2()2(

xPxP

xP

xPxPxPxP


894.)6(.)4(.!0 !8

!8)6(.)4(.

!1 !7

!81

)0()1(1

)1(1

)8(...)3()2()2(

8071

xPxP

xP

xPxPxPxP


Table 4.4

How to use the Binomial Tables

•First find the appropriate table for the particular value of n

•then find the value of p in the top row

•Find the row corresponding to k and find the intersection with the column corresponding to the value of p

•The value you obtain is the cumulative probability, that is P(x ≤ k)

•N=10, p = 0.7: P(x ≤ 4) = 0.047

•N=10, p = 0.7: P(x = 4) = P(x ≤ 4) - P(x ≤ 3) = 0.047-0.011=0.036

•N=10, p = 0.7: P(x > 4) = 1- P(x ≤ 4)

= 1 - 0.047 = 0.953


)1(1

)8(...)3()2()2(

xP

xPxPxPxP


894.

106.01

)1(1

)8(...)3()2()2(

xP

xPxPxPxP

pq

npqnp

1

Mean and Standard deviation

Keys to success

Learn the binomial table.

Be able to recognize binomial distributions and when you do apply the appropriate formulas and tables.

chapter 4. discrete random variables a random variable is a way of recording a quantitative variable...

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