Chapter 24.   Binomial, Poisson, and Hypergeometric Distributions


2018학년도 2학기

       반도체 공학과 공학수학2 (GEDB005) 강의교안  (2018학년도용)

    주교재: Erwin Kreyszig, Engineering Mathematics10th Edition

    부교재: 최신공학수학 I 과 II, 한빛출판사 및 (영문) 다변량 Calculus (이상구, 김응기 et al)   (http://www.hanbit.co.kr/EM/sage/)

    강의시간: E12 (금 12:00 - 14:45), 반도체관 400126호 담당교수: 김응기 박사 

 

주차

주교재

부교재

12

24.7: Binomial, Poisson, and Hypergeometric Distributions

https://youtu.be/uzkc-qNVoOk 

https://youtu.be/iG995W0XefU 

https://youtu.be/peEsXbdMY_4 

web

http://matrix.skku.ac.kr/2018-album/R-Sage-Stat-Lab-2.html 

 


   Week 12



24.7 Discrete Probability Distribution


A discrete probability distribution can be encoded by a discrete list of the probabilities of the outcomes, known as a probability mass function. 



1. Bernoulli Distribution 


Assumptions of Bernoulli Trials. There are three:

Independent repeated trials of an experiment with exactly two possible outcomes are called Bernoulli trials.

① Call one of the outcomes "success" and the other outcome "failure".

② Let   be the probability of success in a Bernoulli trial, and  be the probability of failure.

③ Each events are independent.


If a discrete r.v. , takes only two values (success) and (failure) with probabilities  and  such that

      ,    where 

The distribution is Bernoulli distribution, denoted .


The mean and variance of the Bernoulli r.v. X are

   

   


Example

Show that the Bernoulli distribution is a probability function.

Solution

 Bernoulli distribution is a probability function.



Example

Let a r.v.  denote the outcome of throwing a fair die.

When "1" appears, r.v  has a value of = 1, and when another value comes, . Find the probability distribution of .

Solution

Bernoulli distribution of  is

     


Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

[5/6, 1/6]


R Coding in SageCell

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

[1] 0.8333333 0.1666667




2. Binomial Distribution 


We consider event  and  such that . The probability of exactly  successes in  repeated event  is

         .

Here,  is the binomial coefficient.


     

For ,   is probability mass function with .



The mean and variance of the binomial distribution are

   

   



Shape of binomial distribution

The form of the binomial distribution is determined by the parameters  and 

(1)       symmetric

(2)       Tilted to the left

(3)       Tilted to the right



Example 

Show that binomial distribution is a probability mass function.

Solution

   .



Example

Compute the probability of obtaining at least two “” in rolling a fair die 4 times.

Solution

   .

The event “At least two ‘’” occurs if we obtain  or  or  “.”

Answer is

   

    


Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

19/144



R Coding in SageCell

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

0.1319444



Example

Let  be the number of heads (successes) in  independent tosses of an unbiased coin. The pmf of  is

.

Find  and .

Solution

Let  and 


Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

4

2

37/256

7/64


R Coding in SageCell

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

[1] 4

[1] 2

[1] 0.1445313

[1] 0.109375



Example

Team  has probability  of winning whenever it plays. If  plays 4 games, find the probability that  wins (i) exactly  games (ii) at least  games (iii) more than half of the games.

Solution

Let  and .

(i) The probability of  wins

   

(ii) The probability of  loses 

    The probability of winning at least  game is

      

(iii) The probability of  win or  win are

   .


Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

8/27

80/81

16/27


R Coding in SageCell

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

[1] 0.2962963

[1] 0.9876543

[1] 0.5925926



Example

Let a r.v.  denote the outcome of throwing a fair die. Find the mean and variance of .

Solution

The die is fair, the pmf of  is


Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

7/2

91/6

35/12



Example

Let  be a binomial distributed random variable with  and . Find distribution of .

Solution

           

 

   

 

   

 


Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

[64/729, 64/243, 80/243, 160/729, 20/243, 4/243, 1/729]



R Coding in SageCell

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

[1] 0.087791495 0.263374486 0.329218107 0.219478738 0.082304527 0.016460905 0.001371742




3. Multinomial Distribution 


The binomial distribution is generalized as follows. Suppose the sample space of an experiment is partition into, say,  mutually exclusive events , with respective probabilities .


In  repeated events, the probability that  occurs  times,  occurs  times, , and  occurs  times is equal to

      

(where, ).

The terms in the expansion of  is

      .

If  then we obtain the binomial distribution.



Example

A fair die is tossed  times. The probability of obtaining the faces  and  twice and each of the others once is

      .


Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

35/5832


R Coding in SageCell

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

[1] 0.006001372




4. Hypergeometric Distribution 


Sampling without replacement means that we return no screw to the box. Then we no longer have independence of trials, and instead of binomial distribution the probability of drawing  defectives in  trials is

        ()


(a)  different ways of picking  things from ,

(b)  different ways of picking  defectives from ,

(c)   different ways of picking nondefectives from 

and each way in (b) combined with each way in (c) gives the total number of mutually exclusive ways of obtaining  defectives in  drawings without replacement. Since (a) is the total number of outcomes and we draw at random, each such way has the probability 



The mean and variance of the hypergeometric r.v. X are

   

   

 is a finite population correction coefficient,  as .



Example 

Show that Hypergeomeric distribution is a probability mass function.

Solution

             

 is a probability mass function.

Example

A deck of cards contains  cards:  red cards and  black cards.  cards are drawn randomly without replacement. What is the probability that exactly  red cards are drawn?

Solution

 is red,  is black,

For sampling without replacement.

The probability of red  is



Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

5/21


R Coding in SageCell

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

[1] 0.2380952


Example

We want to draw random samples of two gaskets from a box containing  gaskets, three of which are defective. Find the probability function of the random variable  Number of defective in the sample.

Solution

We have .

For sampling with replacement,

   

For sampling without replacement,

   


Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

[7/15, 7/15, 1/15]


R Coding in SageCell

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

[1] 0.46666667 0.46666667 0.06666667


Derivation of Binomial Distribution from Hypergeometric Distribution

Show that the binomial distribution can be used as a convenient approximation to the Hypergeometric distribution for large  and small .

             

 

 

          

  

  

  

 


5. Poisson Distribution 


A r.v.  is a Poisson distribution r.v. with parameter (in ) if it pmf is

       ().


The cdf of  of Poisson distribution is

         .


The mean and variance of the Poisson r.v. X are

   

   



Example

Show that the Poisson distribution is a probability function.

Solution

 ().

 is pmf.

 Poisson distribution is a probability function.



Example

Suppose that  has a Poisson distribution with . Then the p.m.f, of  is

      ,   

The variance of this distribution is .

Compute .

Solution

      


Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

-e^(-3) + 1


R Coding in SageCell

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

[1] 0.9502129


Example

If the probability of producing a defective screw is  what is the probability that a lot of 100 screws will contain more than 2 defectives?

Solution

.

From the binomial distribution with mean 

         

          


Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

-5/2*e^(-1) + 1


R Coding in SageCell

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

[1] 0.0803014



Example

If the random variable  has a Poisson distribution such that , find .

Solution

            

            


Sage Coding

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

[mu == 0, mu == 2]




Evaluate

2/3*e^(-2)


R Coding in SageCell

http://math3.skku.ac.kr/  http://sage.skku.edu/  http://mathlab.knou.ac.kr:8080/




Evaluate

[1] 0.09022352



Derivation of the Poisson distribution in binomial distributions

Show that the Poisson distribution can be used as a convenient approximation to the binomial distribution for large  and small .

              

              

 

 

         

     



   http://matrix.skku.ac.kr/sglee/ 


[한빛 아카데미] Engineering Mathematics with Sage:

[저자] 이상 구, 김영 록, 박준 현, 김응 기, 이재 화


Contents

 A. 공학수학 1 – 선형대수, 상미분방정식+ Lab

Chapter 01 벡터와 선형대수 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-1.html 


Chapter 02 미분방정식의 이해 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-2.html 

Chapter 03 1계 상미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-3.html    

Chapter 04 2계 상미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-4.html 

Chapter 05 고계 상미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-5.html 

Chapter 06 연립미분방정식, 비선형미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-6.html 

Chapter 07 상미분방정식의 급수해법, 특수함수 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-7.html  

Chapter 08 라플라스 변환 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-8.html 

 

B. 공학수학 2 - 벡터미적분, 복소해석 + Lab

Chapter 09 벡터미분, 기울기, 발산, 회전 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-9.html

Chapter 10 벡터적분, 적분정리 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-10.html

Chapter 11 푸리에 급수, 적분 및 변환 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-11.html

Chapter 12 편미분방정식 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-12.html

Chapter 13 복소수와 복소함수, 복소미분 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-13.html

Chapter 14 복소적분 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-14.html

Chapter 15 급수, 유수 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-15.html

Chapter 16 등각사상 http://matrix.skku.ac.kr/EM-Sage/E-Math-Chapter-16.html

 





Copyright @ 2018 SKKU Matrix Lab. All rights reserved.
Made by Manager: Prof. Sang-Gu Lee and Dr. Jae Hwa Lee
*This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A1B03035865).