Chapter 4. Discrete Random Variables A random variable is a way of recording a quantitative variable...
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Transcript of Chapter 4. Discrete Random Variables A random variable is a way of recording a quantitative variable...
![Page 1: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/1.jpg)
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
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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
![Page 3: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/3.jpg)
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.
![Page 4: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/4.jpg)
Properties of Probability, P( X = xi )
1)(0 (1) ixXP
1)( (2)1
n
iixXP
![Page 5: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/5.jpg)
Copyright © 2013 Pearson Education, Inc.. All rights reserved.
Definition
![Page 6: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/6.jpg)
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.
![Page 7: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/7.jpg)
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.
![Page 8: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/8.jpg)
Copyright © 2013 Pearson Education, Inc.. All rights reserved.
Probability distribution for a two-coin toss
![Page 9: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/9.jpg)
Copyright © 2013 Pearson Education, Inc.. All rights reserved.
Definition
![Page 10: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/10.jpg)
Copyright © 2013 Pearson Education, Inc.. All rights reserved.
Definition
![Page 11: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/11.jpg)
Copyright © 2013 Pearson Education, Inc.. All rights reserved.
Definition
![Page 12: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/12.jpg)
Copyright © 2013 Pearson Education, Inc.. All rights reserved.
Procedure
![Page 13: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/13.jpg)
Copyright © 2013 Pearson Education, Inc.. All rights reserved.
Figure 4.6 Shapes of two probability distributions for a discrete random variable x
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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
![Page 15: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/15.jpg)
Copyright © 2013 Pearson Education, Inc.. All rights reserved.
Graph of p(x)
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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
![Page 17: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/17.jpg)
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?
![Page 18: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/18.jpg)
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.
![Page 19: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/19.jpg)
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
![Page 20: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/20.jpg)
Factorials
0! = 1
1! = 1
2! = 2 * 1
3! = 3 * 2 * 1
4! = 4* 3 * 2 * 1
n! = n*(n-1)!
![Page 21: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/21.jpg)
Factorials
0! = 1
1! = 1
2! = 2 * 1=2
3! = 3 * 2 * 1=6
4! = 4* 3 * 2 * 1=24
n! = n*(n-1)!
![Page 22: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/22.jpg)
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
!)!(
!) (
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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?
![Page 24: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/24.jpg)
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
![Page 25: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/25.jpg)
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.
![Page 26: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/26.jpg)
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.
)8(...)3()2()2( xPxPxPxP
![Page 27: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/27.jpg)
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.
)1(1
)8(...)3()2()2(
xP
xPxPxPxP
![Page 28: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/28.jpg)
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.
)0()1(1
)1(1
)8(...)3()2()2(
xPxP
xP
xPxPxPxP
![Page 29: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/29.jpg)
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
![Page 30: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/30.jpg)
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.
894.)6(.)4(.!0 !8
!8)6(.)4(.
!1 !7
!81
)0()1(1
)1(1
)8(...)3()2()2(
8071
xPxP
xP
xPxPxPxP
![Page 31: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/31.jpg)
Copyright © 2013 Pearson Education, Inc.. All rights reserved.
Table 4.4
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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
![Page 33: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/33.jpg)
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.
![Page 34: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/34.jpg)
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.
)1(1
)8(...)3()2()2(
xP
xPxPxPxP
![Page 35: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/35.jpg)
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.
894.
106.01
)1(1
)8(...)3()2()2(
xP
xPxPxPxP
![Page 36: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/36.jpg)
pq
npqnp
1
Mean and Standard deviation
![Page 37: 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.](https://reader038.fdocuments.in/reader038/viewer/2022102506/56649efc5503460f94c10642/html5/thumbnails/37.jpg)
Keys to success
Learn the binomial table.
Be able to recognize binomial distributions and when you do apply the appropriate formulas and tables.