Chapter 4-2 Continuous Random Variables 主講人 : 虞台文

Content Functions of Single Continuous Random Variable Jointly Distributed Random Variables Independence of Random Variables Distribution of Sums Distributions of Multiplications and Quotients Conditional Densities Multivariate Distributions Multidimensional Changes of Variables

Functions of Single Continuous Random Variable Chapter 4-2 Continuous Random Variables

The Problem 已知 =?=?

Example 11 已知 =?=?

Example 11

Example 12

請熟記 ! 標準常態之平方為一個自由度的卡方

Example 13  0 as 0  y <16  0 as 0  y  4

Example 13  0 as 0  z <4  0 as 0  z  2

Example 14 x fX(x)fX(x) 1

x fX(x)fX(x) 1 How to generate exponentially distributed random numbers using a computer?

Example 14 x fX(x)fX(x) 1

Theorem 1 Let g be a differentiable monotone function on an interval I, and let g(I) denote its range. Let X be a continuous r. v. with pdf f X such that f X (x) = 0 for x  I. Then, Y = g(X) has pdf f Y given by g or

Theorem 1 Let g be a differentiable monotone function on an interval I, and let g(I) denote its range. Let X be a continuous r. v. with pdf f X such that f X (x) = 0 for x  I. Then, Y = g(X) has pdf f Y given by g or

X Y = g(X) Theorem 1 Y = g (X) and Case 1: Pf) y g1(y)g1(y) positive

X Y = g(X) Theorem 1 Y = g (X) and Case 2: Pf) y g1(y)g1(y) negative

Example 15 Redo Example 14 using Theorem 1. x fX(x)fX(x) 1 1 Y = g (X) and

Example 16 Y = g (X) and

Example 16 Y = g (X) and

Example 17

 < 1 : DFR  = 1 : CFR  > 1 : IFR

Example 18 Let X be a continuous random variable. Define Y to be the cdf of X, i.e., Y = F X (X). Find f Y (y).

Random Number Generation The method to generate a random number X such that it possesses a particular distribution by a computer: 1. Let Y = F X (X). 2. Find 3. Generate a random variable by a computer in interval (0, 1). Let y be such a random number. 4. Computing, we obtain the desired random number x.

Example 19 How to generate a random variable X by a computer such that X ~ Exp( )? Let Y = F X (X) = 1  e  X. So, Y ~ U(0, 1). Assume U(0, 1) can be generated by a computer. By letting X =   1 ln(1  Y), we then have X ~ Exp( ).

Jointly Distributed Random Variables Chapter 4-2 Continuous Random Variables

Definition  Joint Distribution Functions The joint (cumulative) distribution function (jcdf) of random variables X and Y is defined by: F X,Y (x, y) = P(X  x, Y  y),  < x < ,  < y < . (x, y)

Properties of a jcdf (x 1, y 1 ) (x 2, y 2 )

Properties of a jcdf ab c d (b, d) (b, c) (a, d) (a, c)

Definition  Marginal Distribution Functions Given the jpdf F(x, y) of random variables X, Y. The marginal distribution functions of X and Y are defined respectively by

Definition  Joint Probability Density Functions A joint probability density function (jpdf) of continuous random variable X, Y is a nonnegative function f X,Y (x, y) such that

Properties of a Jpdf fX(u)fX(u) fY(v)fY(v)

fX(u)fX(u) fY(v)fY(v) Marginal Probability Density Functions (see next page) Marginal Probability Density Functions (see next page)

Marginal Probability Density Functions

Example 20

X Y

X Y (x, y)

Example 20 X Y (x, y)

Example 20 X Y (x, y)

Example 20

Example 21

X Y fX(x)fX(x) x y=xy=x

X Y fY(y)fY(y) y x=yx=y

X Y 105 5

Independence of Random Variables Chapter 4-2 Continuous Random Variables

Definition  Independence of Random Variables Two random variables X and Y are said to be independent, denoted as, if

Theorem 2

Example 22 ?

?

? 

Example 23 ?

? 1 1

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Distribution of Sums Chapter 4-2 Continuous Random Variables

The Problem Given f(x, y) or F(x, y), and Z =  (X, Y), F Z (z) = ? f Z (z) = ?

The Problem Given f(x, y) or F(x, y), and Z =  (X, Y), F Z (z) = ? f Z (z) = ? x y AzAz

The Distribution of Sums Given f(x, y) or F(x, y), and Z =  (X, Y), F Z (z) = ? f Z (z) = ? x y X + Y

The Distribution of Sums x y Z = X + Y

The Distribution of Sums Z = X + Y

The Distribution of Sums Z = X + Y

The Distribution of Sums Z = X + Y I(X), I(Y)  0 and

Example 24 Let X ~ Exp( ), and Y ~ Exp( ) be independent. Let Z = X + Y. Find f Z (z). I ( X ), I ( Y )  0 Fact:

Example 25 Let X ~ U(0, 1) and Y ~ U(0, 1) be two independent variables. Find f X+Y. X ~ U(0, 1), Y ~ U(0, 1) f X+Y = ?

Example 25 X ~ U(0, 1), Y ~ U(0, 1) f X+Y = ? I(X), I(Y)  0 Define 積分區間分析

Example 25 X ~ U(0, 1), Y ~ U(0, 1) f X+Y = ? 積分區間分析 Case 1:Case 2: 0 1 z1z1 z 0 1 z1z1 z I(X), I(Y)  0 Define 0< z  1 1 < z < 2

Example 25 X ~ U(0, 1), Y ~ U(0, 1) f X+Y = ? Case 1:Case 2: I(X), I(Y)  0 Define 0< z  1 1 < z < 2

Example 25 X ~ U(0, 1), Y ~ U(0, 1) f X+Y = ? Case 1:Case 2: I(X), I(Y)  0 Define 0< z  1 1 < z < 2 z fZ(z)fZ(z)

Example 26 x y f(x, y) X ~ U(0, 1), Y ~ U(0, 1)

Example 26 x y f(x, y) X ~ U(0, 1), Y ~ U(0, 1) x y 2x + y = 2

Example 26 x y f(x, y) X ~ U(0, 1), Y ~ U(0, 1) x y x  y =  0.5 x  y = 0.5

Distributions of Multiplications and Quotients Chapter 4-2 Continuous Random Variables

Distributions of Multiplications and and Quotients 已知 I(X), I(Y)  0 and

Example 27 Let X ~  (  1, ) and Y ~  (  2, ) be independent random variables. Find the pdf of Y/X.

Example 27 Chapter 2 Exercise

Example 27

Conditional Densities Chapter 4-2 Continuous Random Variables

Conditional Densities Let X and Y be continuous random variables having jpdf f. The conditional density f Y|X is defined by

Facts

Example 28

Example 29

Multivariate Distributions Chapter 4-2 Continuous Random Variables

Definitions

Properties of Multivariate Distributions

Definition  Independence Random variables X 1, …, X n are called independent if

Example 30

Example 31

Important Theorem of Sums To be proved in the next chapter.

Important Theorem of Sums

熟記 !!! 靈活的將它們用於解題

Multidimensional Changes of Variables Chapter 4-2 Continuous Random Variables

Multidimensional Changes of Variables Let X 1, X 2, …, X n be continuous r.v.’s with jpdf

Multidimensional Changes of Variables Let X 1, X 2, …, X n be continuous r.v.’s with jpdf 假設此函式為一對一 求反函式

Multidimensional Changes of Variables Let X 1, X 2, …, X n be continuous r.v.’s with jpdf 一對一 求反函式 Jacobin Matrix Jacobin Matrix

Example 34

求反函式 Jacobin Matrix Jacobin Matrix

Example 34 求反函式 Jacobin Matrix Jacobin Matrix

Example 34 求反函式 Jacobin Matrix Jacobin Matrix

Example 35

Define ? 求反函式 Jacobin Matrix Jacobin Matrix

Example 35 Define 求反函式 Jacobin Matrix Jacobin Matrix ?

Example 35 Define 求反函式 Jacobin Matrix Jacobin Matrix

Example 35 Define 求反函式 Jacobin Matrix Jacobin Matrix

Example 36 此例非一對一，以上方法非直接可用， 請參考講義。