Chapter 24. Continuous 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 |
13 |
24.8: Standard deviation and Normal Distribution 24.9: Distributions of Several Random Variables |
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web |
Week 13
24.8 Continuous Probability Distributions
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/
Evaluate
3/10
2/5
1/2
R Coding in SageCell
http://math3.skku.ac.kr/ http://sage.skku.edu/ http://mathlab.knou.ac.kr:8080/
Evaluate
[1] 0.3
[1] 0.4
[1] 0.5
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/
Evaluate
1/10*s
0
1
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)
Sage Coding
http://math3.skku.ac.kr/ http://sage.skku.edu/ http://mathlab.knou.ac.kr:8080/
Evaluate
0.997300203937
0.726275075847
R Coding in SageCell
http://math3.skku.ac.kr/ http://sage.skku.edu/ http://mathlab.knou.ac.kr:8080/
Evaluate
[1] 0.9973002
[1] 0.7262751
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/
Evaluate
0.438219823288
0.728667878107
0.0375379803485
0.890651447574
R Coding in SageCell
http://math3.skku.ac.kr/ http://sage.skku.edu/ http://mathlab.knou.ac.kr:8080/
Evaluate
[1] 0.4382198
[1] 0.7286679
[1] 0.03753798
[1] 0.8906514
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/
Evaluate
0.758036347777
R Coding in SageCell
http://math3.skku.ac.kr/ http://sage.skku.edu/ http://mathlab.knou.ac.kr:8080/
Evaluate
[1] 0.7580363
[1] 8.919928
[1] 0.1586553
Evaluate
[1] 7.740557
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).