Chapter 24. Probability & Distributions
2018학년도 2학기
반도체 공학과 공학수학2 (GEDB005) 강의교안 (2018학년도용)
주교재: Erwin Kreyszig, Engineering Mathematics, 10th 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 |
11 |
(newly added) Conditional Probability and Independence, Bayes' theorem 24.5: Random Variables. Probability Distributions 24.6: Mean and Variance of a Distribution |
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Week 11
(newly added)
Conditional Probability and Independence
1. Conditional Probability
Let be an arbitrary event in sample space with . The conditional probability of given , written , is defined as follow :
(where, )
Let be an event for . The conditional probability satisfies the axioms of probability space;
① For any event ,
② If , then
③
Example
Find ,
(1) (2)
Solution
(1)
(2)
(a) (b)
Example
In a certain college, of students failed calculus, of students failed engineering mathematics and of students failed both calculus and engineering mathematics. A student is selected at random.
(i) If he failed engineering mathematics, what is the probability that he failed calculus?
(ii) If he failed calculus, what is the probability that he failed engineering mathematics?
(ii) what is the probability that he failed calculus or engineering mathematics?
Solution
and
, ,
(i) The probability that a student failed calculus, given that he has failed engineering mathematics is
(ii) The probability that a student failed engineering mathematics, given that he has failed calculus is
(iii)
Sage Coding
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Evaluate
1/2
2/5
2. Multiplication Theorem for Conditional Probability
The multiplicative theorem can be obtained from the definition of conditional probability.
In general, the following is satisfied for event , , , ;
(a) Independent Events
Two events and are said to be mutually independent if and only if
.
Example
If and are independent events, then and are independent events.
Proof
Example
A woman is dealt cards one after the other from an ordinary desk of cards. What is the probability that they are all clovers?
Solution
The first card is a clover is .
The second card is a clover is .
The third card is a clover is .
Sage Coding
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Evaluate
11/850
(b) Independent Events and Dependent Events
If and are independent events, then .
If and are dependent events, then .
Example
A box contains screws, three of which are defective. Two screws are drawn at random. Find the probability that neither of the two screws is defective.
Solution
We consider the events
: First drawn screw nondefective.
: Second drawn screw nondefective.
The events are independent, and the answer is
,
If we sample without replacement, then . If has occurred, then there are screws left in the box, of which are defective. Thus and Multiplication Theorem yields the answer
Sage Coding
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Evaluate
49/100
Evaluate
7/15
3. Partition
A partition of a set is a set of nonempty subsets of such that every element in is in exactly one of these subsets.
Equivalently, a family of sets is a partition of if and only if all of the following conditions hold:
①
②
③
Suppose the events , , , from a partition of a sample space .
4. Total probability
Suppose the events , , , is a partition of a sample space . The events are mutually exclusive and their union is . Let be any other event. Then
where, are mutually exclusive. Accordingly,
.
On the other hand, the total probability is obtained by the multiplication theorem of probability
Also, for any , the conditional probability of , given is defined by
The following formula of total probability
If is used, the following equation
where,
is called Bayes theorem.
In the Bayes theorem, is the priori probability of event , and is the posteriori probability of event .
5. Bayes’ Theorem
Suppose the events , , , is a partition of a sample space and is any event. Then for any
Proof
is mutually exclusive event
Conditional probability for any event
,
.
Example
Three machine , , produce respectively , and of the total number of items of a factory. The percentages of defective output of these machines are , and . An item is selected at random, find the probability that item is found defective and
Solution
is the event that an item is defective.
Sage Coding
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Evaluate
29/1000
9/29
24.5 Random Variable
1. Random Variable
The quantity that we observe in an experiment will be denoted by and called a random variable (or stochastic variable) because the value it will assume in the next trial depends on chance, on randomness.
If we count (cars on a road, defective screws in a production, tosses until a die shows the first Six), we have a discrete random variable and distribution. If we measure (electric voltage, rainfall, hardness of steel), we have a continuous random variable and distribution. Precise definitions follow. In both cases the distribution of X is determined by the distribution function
(1)
this is the probability that in a trial, will assume any value not exceeding .
Definition Random Variable
A random variable is a function defined on the sample space of an experiment. Its values are real numbers. For every number the probability
with which X assumes is defined.
From (1) we obtain the fundamental formula for the probability corresponding to an
interval ,
(2)
This follows because (“ assumes any value not exceeding ”) and (“ assumes any value in the interval ”) are mutually exclusive events, so that
by (1) and
and subtraction of on both sides gives (2).
2. Discrete Random Variable and Probability Distribution
A random variable and its distribution are discrete if assumes only finitely many or at most countably many values , , , called the possible values of , with positive probabilities , , , whereas the probability is zero for any interval containing no possible value.
The discrete distribution of is also determined by the probability function of , defined by
(3) ()
From this we get the values of the distribution function by taking sums,
(4)
where for any given we sum all the probabilities for which is smaller than or equal to that of .
For the probability corresponding to intervals we have from (2) and (4)
(5) ( discrete).
This is the sum of all probabilities for which satisfies . From this and we obtain the following formula.
(6) (sum of all probabilities)
a) Probability Mass Function
Let be a discrete random variable on a sample space with a finite image set; . We make into a probability space by defining the probability of to be which we write . This on , defined by is the probability function or probability mass function of and is given in the form of a table.
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b) Properties of Probability Mass Function
Let be a discrete random variable. The function is the probability mass function or discrete probability function of ; it satisfies the following conditions of
(i) ,
(ii)
(iii)
c) Cumulative Distribution Function
The distribution function[or cumulative distribution function] of is the probability mass function defined by
.
The cdf of a discrete r.v. can be obtained by
is a step function.
d) Cumulative Distribution Function
(1)
(2)
(3)
is the monotone increasing function of
(4) is the right continuous function
(5) If is the discrete type then
Example
In the experiment of tossing a fair coin three times. If is the r.v. giving the number of heads obtained, find probability function and distribution function.
Solution
is head, is a tail.
In three toss, the number of heads are
.
Sample space is .
The probability function of is
The distribution function of is
,
,
,
,
,
The distribution function of is
Sage Coding
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Evaluate
[1/8, 3/8, 3/8, 1/8]
[1/8, 1/2, 7/8, 1]
3. Continuous Random Variable and Probability Distribution
Continuous random variables appear in experiments in which we measure. By definition, a random variable and its distribution are of continuous type or, briefly, continuous, if its distribution function [defined in (1)] can be given by an integral
(7) .
Differentiation gives the relation of to as
(8)
for every at which is continuous.
From (2) and (7) we obtain the very important formula for the probability corresponding to an interval:
(9) .
This is the analog of (5).
From (7) and we also have the analog of (6):
(10) .
a) Probability Density Function
Let be a continuous random variable whose image set; is a continuous of numbers such as an interval. The set is an events in and therefore the probability is
.
is a probability function and probability density function of .
b) Properties of Probability Density Function
Let be a continuous random variable. The function is the probability density function or continuous probability function of ; it satisfies the following conditions
(1) (2) (3)
c) Cumulative Distribution Function
Let be a continuous random variable. The distribution function or cumulative distribution function of is
.
If is a continuous random variable with distribution , then
.
d) Properties of Cumulative Distribution Function
(1) where , is the probability of .
(2) ,
(3) ,
(4) for all
(5)
is the monotone increasing function of
(6)
(7) If is the continuous type then
.
Example
Let be a continuous random variable with probability density function
.
(i) Find
(ii) Find a cumulative distribution function.
Solution
(i)
(ii) The cumulative distribution function is
Sage Coding
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Evaluate
1/4
Evaluate
1/4*s^2
0
1
5. Expected Value
The expected value means average. and are the expected value.
This on , defined by is the probability function or probability mass function of .
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(i) (ii)
The expected value of is
Properties of the Expected Value
(1) (2) (3)
Example
Let have the pmf
then find .
Solution
Sage Coding
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Evaluate
[1/6, 1/6, 1/6, 1/6, 1/6, 1/6]
7/2
Example
In the experiment of tossing a fair coin two times. If is the r.v. giving the number of heads obtained, find and .
Solution
is head, is a tail.
In two toss, the number of heads are
.
Sample space is .
The probability function of is
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The probability mass function of is
Sage Coding
http://math3.skku.ac.kr/ http://sage.skku.edu/ http://mathlab.knou.ac.kr:8080/
Evaluate
[1/4, 1/2, 1/4]
1
3/2
Example
Let have the pdf
then find and .
Solution
Sage Coding
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Evaluate
2
16/3
Variance
The variance of a r.v. denoted by or is defined by
Proof
Remark
The standard deviation of a r.v. denoted by , is the positive square root of .
Properties of the Variance
(1) (2) (3)
Mean and Variance
The mean (or expected value) of a r.v. , denoted by or , is defined by
The variance of a r.v. , denoted by or , is defined by
Example
Let be a continuous random variable with probability density function
where is constant.
(i) Determine .
(ii) Find the mean of
(iii) Find the variance of
Solution
(i)
(ii)
(iii)
Example
Let be a continuous random variable with probability density function
.
Find the mean and the variance of .
Solution
,
Sage Coding
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Evaluate
2/3
1/2
1/18
Evaluate
4
33/2
1/2
Example
Find the mean and variance, if they exist, of each of the following distributions.
(a)
(b)
Solution
(a)
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1/2
3/10
1/20
(b)
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Evaluate
2
6
2
Example
For each of the following distributions, compute .
(a)
(b)
Solution
(a)
Sage Coding
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Evaluate
1/2
3/10
1/20
11/25*sqrt(5)
(b)
(b)
,
Sage Coding
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Evaluate
2
6
2
13/8
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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
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).