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
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
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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
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Example
Let be the number of heads (successes) in independent tosses of an unbiased coin. The pmf of is
.
Find , , and .
Solution
Let and
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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
.
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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
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Example
Let be a binomial distributed random variable with and . Find distribution of .
Solution
,
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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
.
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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
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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,
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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
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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
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Example
If the random variable has a Poisson distribution such that , find .
Solution
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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)
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Example
Let r.v. be . Find the pdf and cdf.
Solution
The pdf of is
The cdf of is
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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.
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 .
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)
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Example
Using the standard normal distribution, find the next probability
(1) (2)
(3) (4)
Solution
(1)
(2)
(3)
(4)
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Example
Let r.v. be , find the following.
(1) Find .
(2) Find in .
(3) Find in .
Solution
(1)
(2)
(3)
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[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).