2019학년도 2학기

                   반도체 공학과 공학수학2


   주교재: Erwin Kreyszig, Engineering Mathematics, 10th Edition

   부교재: 이상구 외 4인, 최신공학수학 Ⅱ   1st Edition  

   실습실: http://www.hanbit.co.kr/EM/sage/

          http://matrix.skku.ac.kr/2018-album/R-Sage-Stat-Lab-1.html  

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

   강의시간:  공학수학2, (화16:30-17:45)   (목15:00-16:15)

   담당교수:  김응기 박사



Week 15


15주차

24.7: Binomial, Poisson, and Hypergeometric Distributions

  https://youtu.be/uzkc-qNVoOk    https://youtu.be/iG995W0XefU 

  https://youtu.be/peEsXbdMY_4 

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


24.8: Normal Distribution

  https://youtu.be/0ZstEh_8bYc    https://youtu.be/JNm3M9cqWyc 

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

  http://matrix.skku.ac.kr/2018-EM/EM-2-W12-lab.html 

  http://matrix.skku.ac.kr/2018-EM/EM-2-W13-lab.html 

  http://matrix.skku.ac.kr/2018-EM/EM-2-W14-lab.html 




24.7. Binomial, Poisson, and Hypergeometric

Distributions


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/


R Coding in SageCell

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



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/



R Coding in SageCell

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



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/


R Coding in SageCell

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



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/


R Coding in SageCell

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



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/



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/



R Coding in SageCell

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



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/


R Coding in SageCell

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



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/


R Coding in SageCell

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


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/


R Coding in SageCell

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


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/


R Coding in SageCell

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


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/


R Coding in SageCell

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



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/



R Coding in SageCell

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



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 .

    

                   

                   

     

     

             

         

    



24.8. Normal Distribution


In this chapter we will focus on continuous random variables, cumulative distribution functions and probability density functions of continuous random variables, expected value, variance, and standard deviation of continuous random variables, and some special continuous distributions.



1. Uniformly Distribution


The probability density function for the continuous uniform distribution on the interval is

    .


The Continuous Density Function (CDF) can be derived from the above PDF using the relation

    .


The cumulative distribution function for a continuous uniform distribution on the interval is

    

   

Probability density function and distribution function of Uniformly Distribution

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

    

    



Example

Show that the Uniformly distribution is a probability function.

Solution

    

    

, is pdf.

 Uniformly distribution is a probability function.



Example

Let r.v. be . Find the next probability

(1)                (2)                 (3)

Solution

(1)

(2)

(3)


Sage Coding

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


R Coding in SageCell

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



Example

Let r.v. be . Find the pdf and cdf.

Solution

The pdf of   is

    

The cdf of is

    


Sage Coding

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



2. Normal Distribution


A r.v. is called a normal (or gaussian) r.v. if its pdf is given by

    

 is said to be a normal distribution and denoted by .


The cdf of X is

    .


The graph of and is as follows.

  그림입니다.
원본 그림의 이름: CLP00000f643f62.bmp
원본 그림의 크기: 가로 315pixel, 세로 212pixel  그림입니다.
원본 그림의 이름: CLP00000f640001.bmp
원본 그림의 크기: 가로 318pixel, 세로 220pixel

  Probability density function and distribution function of normal distribution


Properties of the curve of the normal distribution

⑴ The normal curve is bell-shaped.

⑵ The normal curve is symmetrical about the mean .

⑶ The maximum value in is .

⑷ The -axis is the asymptote.  

⑸ The inflection point is , is almost in contact with the -axis.

⑹ The total area under the curve is equal to .



The mean and variance of the normal r.v. are

    

    



Example

Show that the Normal distribution is a probability function.

Solution

(1)

(2)

    

    

          

    

    

    

    

    

 Normal distribution is a probability function.



Example

Let r.v. be . Find the pdf of and pdf of .

Solution

The pdf of

    

The inverse of is ,

The pdf of is

    

   is the normal distribution.



Example

Show that the graph of a pdf has points of inflection at and .

Solution

    

    

    

          

           

   and are a points of inflection of .



3. Standard normal distribution


If we make the substitution in the Normal distribution we obtain the Standard normal distribution of pdf

    

which has mean and variance .

Then, is the standard normal distribution, denoted by .


The standardized probability variable is the standard normal distribution .

The cdf of is

    

 is the standard normal distribution function.



Normalization of Normal Distribution

Let r.v. be the mean and standard variance

    

    


 has a distribution if and only if has a distribution.

                               

                       Distribution of           Distribution of



This gives values of the standard normal distribution at in steps of .

                       그림입니다.
원본 그림의 이름: mem00003598095a.png
원본 그림의 크기: 가로 183pixel, 세로 162pixel

The normal distribution table is a standardized random variable .

(1)            (2)            (3)


Let have a distribution. If we want to compute for

a specified , then for

          

    

   


Probability in normal distribution

    

    

    

    

    

    


Example

(1) Let r.v. be . Find .

(2) Let r.v. be . Find .

Solution

(1)

                  

(2)

                 


Sage Coding

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


R Coding in SageCell

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


Example

Using the standard normal distribution, find the next probability

(1)                    (2)

(3)                      (4)

Solution

(1)

(2)

                      

(3)

(4)

Sage Coding

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


R Coding in SageCell

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


Example

Let r.v. be , find the following.

(1) Find .

(2) Find in .

(3) Find in .

Solution

(1)

(2)            

(3)

                 

                 

                 

    

          


Sage Coding

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


R Coding in SageCell

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




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



Made by Prof. Sang-Gu LEE  sglee at skku.edu

http://matrix.skku.ac.kr/sglee/   with Dr. Jae Hwa LEE


Copyright @ 2019 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).