Chapter 24.   Continuous Probability 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

13

24.8: Standard deviation and Normal Distribution

24.9: Distributions of Several Random Variables

https://youtu.be/0ZstEh_8bYc 

https://youtu.be/JNm3M9cqWyc 

web

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

 


   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.

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 .

                       

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).