Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.
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Transcript of Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.
![Page 1: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/1.jpg)
Chapter 4-2Continuous Random Variables
主講人 :虞台文
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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
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Functions of Single Continuous Random Variable
Chapter 4-2Continuous Random Variables
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The Problem
( )Y g X( )Xf x 已知
( )Yf y =?
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Example 11
2Y X( )Xf x 已知
( )Yf y =?
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Example 112Y X
( )Xf x 已知
( )Yf y =?2Y X
( )Xf x 已知
( )Yf y =?
( ) ?I Y [0, )
( ) ( )YF y P Y y 2( )P X y P y X y
X XF y F y
( ) ( )dY Ydyf y F y d
X Xdy F y F y
d yd yX Xdy dyf y f y
12 X Xy
f y f y 0y
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Example 112Y X
( )Xf x 已知
( )Yf y =?2Y X
( )Xf x 已知
( )Yf y =?
12
( ) , 0Y X Xyf y f y f y y
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Example 12 1
2( ) , 0Y X Xy
f y f y f y y
2Y X
2 , ( ? )YY X f y ~ (0,1).X N2 / 21
2( ) , x
Xf x e x
/ 2 / 21 1 12 2 2
( ) y yY y
f y e e
1
21/ 2 1/ 21
2 , 0yy e
y
2~Y 1 1,
2 2
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Example 12 1
2( ) , 0Y X Xy
f y f y f y y
2Y X
2 , ( ? )YY X f y ~ (0,1).X N
2~Y 1 1,
2 2
請熟記 !
標準常態之平方為一個自由度的卡方
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Example 13
21. , ( ) ?YY X f y
~ ( 2,4).X U
2. | |, ( ? )ZZ X f z
12
( ) , 0Y X Xyf y f y f y y
2Y X
( ) ?I Y [0,16)
12
( )Y X Xyf y f y f y
0as 0 y <16
0as 0 y 4
fX(x)
x 2 4
fX(x)
x 2 4
1/6
fX(x)
x 2 4
fX(x)
x 2 4
1/6
16
112
0 4
4 16
y
y
y
y
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Example 13
21. , ( ) ?YY X f y
~ ( 2,4).X U
2. | |, ( ? )ZZ X f z
( ) ?I Z [0,4)
fX(x)
x 2 4
fX(x)
x 2 4
1/6
fX(x)
x 2 4
fX(x)
x 2 4
1/6
( ) ( )ZF z P Z z (| | )P X z ( )P z X z ( ) ( )X XF z F z
( ) ( )dZ Zdzf z F z ( ) ( )d
X Xdz F z F z ( ) ( )X Xf z f z
0as 0 z <4
0as 0 z 21
3
16
0 2( )
2 4Z
zf z
z
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Example 14
~ (0,1).X U
( ) ?I Y (0, )
( ) ( )YF y P Y y 1( ln(1 ) )P X y
1 ln(1 ) with 0, ( ) ?YY X f y
ln(1 )P X y
(1 )yP X e ( 1)yP X e ( 1 )yP X e
x
fX(x)
1
(1 )yXF e 1 ye
( ) , 0yYf y e y ~ ( )Y Exp
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Example 14 x
fX(x)
1
~ ( )Y Exp
How to generate exponentially
distributed random numbers
using a computer?
How to generate exponentially
distributed random numbers
using a computer?
~ (0,1).X U 1 ln(1 ) with 0, ( ) ?YY X f y
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Example 14 x
fX(x)
1
~ ( )Y Exp
~ (0,1).X U 1 ln(1 ) with 0, ( ) ?YY X f y
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
x
y= 1ln(1-x ) = 2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1
x
y= 1ln(1-x ) = 2 1 ln(1 )Y X
ln(1 )Y X
1Ye X 1 YX e ( )YF Y
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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 fX such that fX(x) = 0 for x I. Then, Y
= g(X) has pdf fY given by
g or
1 1( ) ( ) ( ) , ( )Y X
df y f g y g y y g I
dy
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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 fX such that fX(x) = 0 for x I. Then, Y
= g(X) has pdf fY given by
g or
11( ) , (( ) ( ) )XY
df g y y g I
dyf y g y 11( ) , ( ) ) )( (Y X
d
dyff y yg y gg y I 11( ) , ( ) ) )( (Y X
d
dyff y yg y gg y I 1 1 , ( ) ( ) ( )) (Y X
d
dyff y g y g Ig yy 1 1( ) ( ) ( ) , ( )XY
d
dyff y g y g y y g I
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X
Y = g(X)
Theorem 1
g oror 11( ) , (( ) ( ) )XY
df g y y g I
dyf y g y 11( ) , ( ) ) )( (Y X
d
dyff y yg y gg y I 11( ) , ( ) ) )( (Y X
d
dyff y yg y gg y I 1 1 , ( ) ( ) ( )) (Y X
d
dyff y g y g Ig yy 1 1( ) ( ) ( ) , ( )XY
d
dyff y g y g y y g I Y = g (X) and
Case 1:Pf) g ( ) ( )YF y P Y y ( ( ) )P g X y
y
g1(y)
1( )P X g y 1( )XF g y
1 1( ) ( ) ( )Y X
df y f g y g y
dy
positive
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X
Y = g(X)
Theorem 1
g oror 11( ) , (( ) ( ) )XY
df g y y g I
dyf y g y 11( ) , ( ) ) )( (Y X
d
dyff y yg y gg y I 11( ) , ( ) ) )( (Y X
d
dyff y yg y gg y I 1 1 , ( ) ( ) ( )) (Y X
d
dyff y g y g Ig yy 1 1( ) ( ) ( ) , ( )XY
d
dyff y g y g y y g I Y = g (X) and
Case 2:Pf) ( ) ( )YF y P Y y ( ( ) )P g X y
y
g1(y)
1( )P X g y 11 ( )XF g y
1 1( ) ( ) ( )Y X
df y f g y g y
dy
negative
g
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Example 15Redo Example 14 using Theorem 1.
~ (0,1).X U 1 ln(1 ) with 0, ( ) ?YY X f y
?X Y 1( ) 1 YX g Y e 11. ( ) 1 yg y e
12. ( ) yddy g y e
( ) ?I Y (0, )
x
fX(x)
1
1
13. ( ) yddy g y e
14. ( ) 1Xf g y
g oror 11( ) , (( ) ( ) )XY
df g y y g I
dyf y g y 11( ) , ( ) ) )( (Y X
d
dyff y yg y gg y I 11( ) , ( ) ) )( (Y X
d
dyff y yg y gg y I 1 1 , ( ) ( ) ( )) (Y X
d
dyff y g y g Ig yy 1 1( ) ( ) ( ) , ( )XY
d
dyff y g y g y y g I Y = g (X) and
0, ( )y I Y
(0,1), ( )y I Y
( ) , 0yYf y e y
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Example 16
?X Y 1 2( )X g Y Y 1 21. ( )g y y
12. ( ) 2ddy g y y
( ) ?I Y (0, )
13. ( ) 2ddy g y y
214. ( ) yXf g y e
g oror 11( ) , (( ) ( ) )XY
df g y y g I
dyf y g y 11( ) , ( ) ) )( (Y X
d
dyff y yg y gg y I 11( ) , ( ) ) )( (Y X
d
dyff y yg y gg y I 1 1 , ( ) ( ) ( )) (Y X
d
dyff y g y g Ig yy 1 1( ) ( ) ( ) , ( )XY
d
dyff y g y g y y g I Y = g (X) and
0, ( )y I Y
2
( ) 2 , 0yYf y ye y
1. , ( ) ?YY X f y ~ ( ).X Exp 22. 1 ( ?)ZZ X f z
( ) , 0xXf x e x
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Example 16
?X Z 1( ) 1X g Z Z 11. ( ) 1g z z
1 12 1
2. ( )ddz z
g z
( ) ?I Z ( ,1)
1 12 1
3. ( )ddy z
g z
1 14. ( ) zXf g z e
g oror 11( ) , (( ) ( ) )XY
df g y y g I
dyf y g y 11( ) , ( ) ) )( (Y X
d
dyff y yg y gg y I 11( ) , ( ) ) )( (Y X
d
dyff y yg y gg y I 1 1 , ( ) ( ) ( )) (Y X
d
dyff y g y g Ig yy 1 1( ) ( ) ( ) , ( )XY
d
dyff y g y g y y g I Y = g (X) and
0, ( )z I Z
1
2 1( ) , 1z
Z zf z e z
1. , ( ) ?YY X f y ~ ( ).X Exp 22. 1 ( ?)ZZ X f z
( ) , 0xXf x e x
![Page 22: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/22.jpg)
Example 17
( ) , 0xXf x e x
1/ , 0Y X
![Page 23: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/23.jpg)
Example 17
( ) , 0xXf x e x
1/ , 0Y X
?X Y 1( )X g Y Y 11. ( )g y y
1 12. ( )ddy g y y
( ) ?I Y (0, )
1 13. ( )ddy g y y
14. ( ) yXf g y e
0, ( )y I Y
1( ) , 0yYf y y e y
![Page 24: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/24.jpg)
Example 17
( ) , 0xXf x e x
1/ , 0Y X
1( ) , 0yYf y y e y
1
00
( )0 0
y x
Y
x e dx yF y
y
1 0
0 0
ye y
y
![Page 25: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/25.jpg)
Example 17
( ) , 0xXf x e x
1/ , 0Y X
1( ) , 0yYf y y e y
1 0( )
0 0
y
Y
e yF y
y
( )( )
( )Y
YY
f th t
R t
1 t
t
t e
e
1t
![Page 26: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/26.jpg)
Example 17
( ) , 0xXf x e x
1/ , 0Y X
1( ) , 0Yh t t t
0
0.5
1
1.5
2
2.5
3
3.5
4
0 1 2 3 4 5
t
h (t )
0
0.5
1
1.5
2
2.5
3
3.5
4
0 1 2 3 4 5
t
h (t )
1
0.750.50
0.250.10
1.5
2
2.5
3.01
< 1 : DFR = 1 : CFR > 1 : IFR
![Page 27: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/27.jpg)
Example 18
Let X be a continuous random variable. Define Y to be the cdf of X, i.e., Y = FX(X). Find fY(y).
Let X be a continuous random variable. Define Y to be the cdf of X, i.e., Y = FX(X). Find fY(y).
( ) ?I Y [0,1]
( ) ( )YF y P Y y ( ( ) )XP F X y 1( )XP X F y 1( )X XF F y y
( ) ( )Y Yf y F y 1 0 1y
( )XY F X ~ (0,1)Y U
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Random Number Generation
The method to generate a random number X such that it possesses a particular distribution by a computer:
1. Let Y = FX(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.
( )XY F X ~ (0,1)Y U
1( ).XX F Y
1( )XX F Y
![Page 29: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/29.jpg)
Example 19
How to generate a random variable X by a
computer such that X ~ Exp()?
How to generate a random variable X by a
computer such that X ~ Exp()?
Let Y = FX(X) = 1 eX. So, Y ~ U(0, 1).
Assume U(0, 1) can be generated by a computer.
By letting X = 1ln(1Y), we then have X ~ Exp().
( )XY F X ~ (0,1)Y U
![Page 30: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/30.jpg)
Jointly DistributedRandom Variables
Chapter 4-2Continuous Random Variables
![Page 31: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/31.jpg)
Definition Joint Distribution Functions
The joint (cumulative) distribution function (jcdf) of random variables X and Y is defined by:
FX,Y(x, y) = P(X x, Y y), < x < , < y < .
(x, y)
![Page 32: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/32.jpg)
Properties of a jcdf
1. 0 ( , ) 1, ,F x y x y
1 1 2 2 1 2 1 22. ( , ) ( , ), ,F x y F x y x x y y
(x1, y1)
(x2, y2)
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Properties of a jcdf
1. 0 ( , ) 1, ,F x y x y
1 1 2 2 1 2 1 22. ( , ) ( , ), ,F x y F x y x x y y
, ,3. lim ( , ) 0, lim ( , ) 1
x y x yF x y F x y
4. ( , )P a X b c Y d
a b
c
d(b, d)
(b, c)
(a, d)
(a, c)
( , )F b d ( , )F a d ( , )F b c ( , )F a c
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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
( ) ( , )XF x P X x Y ( , )P X x Y
lim ( , )y
yF x
( ) lim ( , )Y xF y F yx
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Definition Joint Probability Density Functions
A joint probability density function (jpdf) of
continuous random variable X, Y is a nonnegative
function fX,Y(x, y) such that
, ,( ( ), ) ,y x
X XY Yf uF x y dudvv
![Page 36: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/36.jpg)
Properties of a Jpdf
2. ( , ) 1f x y dxdy
3. ( , ) ( , )d b
c aP a X b c Y d f x y dxdy
2,
,
( , )1. ( , ) X Y
X Y
F x yf x y
x y
4. ( ) ( , ) ( ) ( , )x y
X YF x f u y dydu F y f x v dxdv
and
5. ( ) ( , ) ( ) ( , )X Yf x f x y dy f y f x y dx
and
f X(u) f Y(v)
![Page 37: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/37.jpg)
Properties of a Jpdf
2. ( , ) 1f x y dxdy
3. ( , ) ( , )d b
c aP a X b c Y d f x y dxdy
2,
,
( , )1. ( , ) X Y
X Y
F x yf x y
x y
4. ( ) ( , ) ( ) ( , )x y
X YF x f u y dydu F y f x v dxdv
and
5. ( ) ( , ) ( ) ( , )X Yf x f x y dy f y f x y dx
and
f X(u) f Y(v)
Marginal Probability Density Functions(see next page)
Marginal Probability Density Functions(see next page)
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Marginal Probability Density Functions
( ) ( , )
( ) ( , )
X
Y
f x f x y dy
f y f x y dx
![Page 39: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/39.jpg)
Example 20
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Example 20( , ) 1f x y dxdy
1 1
0 01 ( )k x y dxdy 11 21
20 0k x yx dy
1
120
k y dy 121 1
2 2 0k y y k
1k
![Page 41: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/41.jpg)
Example 20
1k
0
0.5
1
1.5
2
0
0.5
1
1.5
2
![Page 42: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/42.jpg)
Example 20
1
0( ) ( )Xf x x y dy
( ) ( , )
( ) ( , )
X
Y
f x f x y dy
f y f x y dx
1k
1
0( ) ( )Yf y x y dx
121
2 0xy y
121
2 0x xy
12x
12y
0 1x
0 1y
![Page 43: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/43.jpg)
Example 2012
12
( ) , 0 1
( ) , 0 1X
Y
f x x x
f y y y
1k
( )XF x 120
0 0
( ) 0 1
1 1
x
x
u du x
x
( )YF y 120
0 0
( ) 0 1
1 1
y
y
v dv y
y
![Page 44: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/44.jpg)
Example 2012
12
( ) , 0 1
( ) , 0 1X
Y
f x x x
f y y y
1k
( )XF x 2
2 2
0 0
0 1
1 1
x x
x
x
x
( )YF y2
2 2
0 0
0 1
1 1
y y
y
y
y
![Page 45: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/45.jpg)
Example 20
, ( , ) , 0 , 1X Yf x y x y x y
1k
, ) ?( ,X YF x y
0
0.5
1
1.5
2
0
0.5
1
1.5
2
, ( , )X Yf x y
![Page 46: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/46.jpg)
Example 20
, ( , ) , 0 , 1X Yf x y x y x y
, ) ?( ,X YF x y
X
Y
1
1
, ( , ) ( , )X YF x y P X x Y y , ( , ) ( , )X YF x y P X x Y y
0
00 0 0
1
![Page 47: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/47.jpg)
Example 20
, ( , ) , 0 , 1X Yf x y x y x y
, ) ?( ,X YF x y
X
Y
1
1
, ( , ) ( , )X YF x y P X x Y y , ( , ) ( , )X YF x y P X x Y y
0
00 0 0
1(x, y)
, 0 0( , ) ( )
y x
X YF x y u v dudv
![Page 48: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/48.jpg)
Example 20
, ( , ) , 0 , 1X Yf x y x y x y
, ) ?( ,X YF x y
X
Y
1
1
, ( , ) ( , )X YF x y P X x Y y , ( , ) ( , )X YF x y P X x Y y
0
00 0 0
1(x, y)
(x, y)
, 0 0( , ) ( )
y x
X YF x y u v dudv 1
, 0 0( , ) ( )
y
X YF x y x v dxdv
![Page 49: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/49.jpg)
Example 20
, ( , ) , 0 , 1X Yf x y x y x y
, ) ?( ,X YF x y
X
Y
1
1
, ( , ) ( , )X YF x y P X x Y y , ( , ) ( , )X YF x y P X x Y y
0
00 0 0
1(x, y)
(x, y)
, 0 0( , ) ( )
y x
X YF x y u v dudv 1
, 0 0( , ) ( )
y
X YF x y x v dxdv
(x, y)1
, 0 0( , ) ( )
x
X YF x y u y dydu
![Page 50: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/50.jpg)
Example 20
, ( , ) , 0 , 1X Yf x y x y x y
, ) ?( ,X YF x y
![Page 51: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/51.jpg)
Example 21
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
, ( , ) , 0xX Yf x y e y x
( ) , ( )? ?X Yf x f y
( 5 1 ?| 0)P Y X
![Page 52: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/52.jpg)
Example 21
, ( , ) , 0xX Yf x y e y x
( ) , ( )? ?X Yf x f y
( 5 1 ?| 0)P Y X
X
Y ( ) ( , )Xf x f x y dy
fX(x)
x
y=x 0
x xe dy, 0xxe x
![Page 53: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/53.jpg)
Example 21
, ( , ) , 0xX Yf x y e y x
( ) , ( )? ?X Yf x f y
( 5 1 ?| 0)P Y X
X
Y ( ) , 0xXf x xe x
fY(y)y
x=y
( ) ( , )Yf y f x y dx
x
ye dx
, 0ye y
![Page 54: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/54.jpg)
Example 21
, ( , ) , 0xX Yf x y e y x
( ) , ( )? ?X Yf x f y
( 5 1 ?| 0)P Y X
X
Y ( ) , 0xXf x xe x
( ) , 0yYf y e y
( 10, 5)( 5 | 10)
( 10)
P X YP Y X
P X
105
5
10
5 510
0
x x
x
e dydx
xe dx
5 10
10
6
1 11
e e
e
0.00647
![Page 55: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/55.jpg)
Independence of Random Variables
Chapter 4-2Continuous Random Variables
![Page 56: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/56.jpg)
Definition Independence of Random Variables
Two random variables X and Y are said
to be independent, denoted as , if X Y
, ( , ) ( ) ( ),
,X Y X YF x y F x F y
x y
![Page 57: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/57.jpg)
Theorem 2
X Y , ( , ) ( ) ( ),
,X Y X YF x y F x F y
x y
X Y , ( , ) ( ) ( ),
,X Y X Yf x y f x f y
x y
, ( , ) ( ) ( ),
,X Y X YF x y F x F y
x y
, ( , ) ( ) ( ),
,X Y X Yf x y f x f y
x y
![Page 58: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/58.jpg)
Example 22
2 2( ) / 2( , )
,
x xy yf x y ce
x y
?
?
?X
Y
c
f
f
X Y ?
![Page 59: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/59.jpg)
Example 22
2 2( ) / 2( , )
,
x xy yf x y ce
x y
?
?
?X
Y
c
f
f
X Y ?
2 2/ 2 3 / 4 / 2( , )
x y yf x y ce
2 2/ 2 3 / 4 / 2( )
x y y
Yf y c e dx
22 / 2 / 23 /8 x yyce e dx
2
23 /82 yc e 2
22 4/3
2
y
c e
12
2 4 / 3c
3
4c
2
22 4/33
2 2
y
e
![Page 60: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/60.jpg)
Example 22
2 2( ) / 2( , )
,
x xy yf x y ce
x y
?
?
?X
Y
c
f
f
X Y ?
2 2/ 2 3 / 4 / 2( , )
x y yf x y ce
2
22 4/33
( ) , 2 2
y
Yf y e y
2
22 4/33
( ) , 2 2
x
Xf x e x
( ) ( )X Xf x f y
![Page 61: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/61.jpg)
Example 23
1 2( )1 2( , ) , 0 ,x yf x y e x y
X Y ?
?( , )F x y
![Page 62: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/62.jpg)
Example 23
1 2( )1 2( , ) , 0 ,x yf x y e x y
X Y ?
?( , )F x y
1 2( )1 20
( ) x yXf x e dy
1 21 20
x ye e dy
1 2( )1 20
( ) x yYf y e dx
2 12 10
y xe e dx
1
1
![Page 63: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/63.jpg)
Example 23
1 2( )1 2( , ) , 0 ,x yf x y e x y
X Y ?
?( , )F x y
11( ) , 0x
Xf x e x
22( ) , 0x
Yf y e y
( , ) ( ) ( )X Yf x y f x f y
1~ ( )X Exp
2~ ( )Y Exp ( , ) ( ) ( )X YF x y F x F y
1 21 1
0 ,
x ye e
x y
![Page 64: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/64.jpg)
Distribution of Sums
Chapter 4-2Continuous Random Variables
![Page 65: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/65.jpg)
The Problem
Given f(x, y) or F(x, y), and Z = (X, Y),
FZ(z) = ?
fZ(z) = ?
![Page 66: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/66.jpg)
The Problem
Given f(x, y) or F(x, y), and Z = (X, Y),
FZ(z) = ?
fZ(z) = ?
x
y
Az
( ) ( )
( , )z
Z
A
F z P Z z
f x y dxdy
( ) ( , ) : ( , )zA z x y x y z
![Page 67: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/67.jpg)
The Distribution of Sums
Given f(x, y) or F(x, y), and Z = (X, Y),
FZ(z) = ?
fZ(z) = ?
x
y
( ) ( )
( , )z
Z
A
F z P Z z
f x y dxdy
( ) ( , ) :z yA z x y x z
X + Y
xz
y
x
zy
![Page 68: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/68.jpg)
The Distribution of Sums
x
y
( ) ( )
( , )z
Z
A
F z P Z z
f x y dxdy
xz
y
x
zy
Z = X + Y
( ,( ))z x
Z f x y dyF z dx
![Page 69: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/69.jpg)
The Distribution of Sums
Z = X + Y
( ,( ))z x
Z f x y dyF z dx
( , )
y z x
yy dxyx df
y t x ( , )t x z x
t xd dt tf x xx
( , )z
dxdt x tf x
( , )z
f x t x dxdt
( )z
Zf t dt
![Page 70: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/70.jpg)
The Distribution of Sums
Z = X + Y
( ,( ))z x
Z f x y dyF z dx
( , )
y z x
yy dxyx df
y t x ( , )t x z x
t xd dt tf x xx
( , )z
dxdt x tf x
( , )z
f x t x dxdt
( )z
Zf t dt
( ) ( , )Zf z f x z x dx
( , )f z y y dy
![Page 71: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/71.jpg)
The Distribution of Sums
Z = X + Y
( ) ( , )Zf z f x z x dx
I(X), I(Y) 00
( ) ( , )z
Zf z f x z x dx
X Y0
( ) ( ) ( )z
Z X Yf z f x f z x dx and
![Page 72: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/72.jpg)
Example 24
Let X ~ Exp(), and Y ~ Exp() be independent. Let Z = X + Y. Find fZ(z).
Let X ~ Exp(), and Y ~ Exp() be independent. Let Z = X + Y. Find fZ(z).
2 ( )( , ) , 0 ,x yf x y e x y X YI(X), I(Y) 0
0( ) ( , )
z
Zf z f x z x dx
2
0( )
z zzf z e dx 2 , 0zze z
Fact: ~ (2, )X Y
![Page 73: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/73.jpg)
Example 25
Let X ~ U(0, 1) and Y ~ U(0, 1) be two independent variables. Find fX+Y.
Let X ~ U(0, 1) and Y ~ U(0, 1) be two independent variables. Find fX+Y.
X ~ U(0, 1), Y ~ U(0, 1) X Y
fX+Y = ?
![Page 74: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/74.jpg)
Example 25X ~ U(0, 1), Y ~ U(0, 1) X Y
fX+Y = ?
( , ) 1, 0 , 1f x y x y I(X), I(Y) 0
.Z X Y Define ( ) ?I Z (0,2)
[0 1][0 1]I x z x 積分區間分析
0( ) ( , )
z
Zf z f x z x dx
[0 1][ 1 ]x z x z [0 1][ 1 ]I x z x z
![Page 75: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/75.jpg)
Example 25X ~ U(0, 1), Y ~ U(0, 1) X Y
fX+Y = ?
[0 1][ 1 ]I x z x z 積分區間分析
Case 1: Case 2:
0 1
z1 z
0 1
z1 z
( , ) 1, 0 , 1f x y x y I(X), I(Y) 0
.Z X Y Define ( ) ?I Z (0,2)0
( ) ( , )z
Zf z f x z x dx 0< z 1 1 < z < 2
[0 ]I x z [ 1 1]I z x
![Page 76: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/76.jpg)
Example 25X ~ U(0, 1), Y ~ U(0, 1) X Y
fX+Y = ?
Case 1: Case 2:
( , ) 1, 0 , 1f x y x y I(X), I(Y) 0
.Z X Y Define ( ) ?I Z (0,2)0
( ) ( , )z
Zf z f x z x dx 0< z 1 1 < z < 2
[0 ]I x z [ 1 1]I z x
0( )
z
Zf z dx z1
1( )Z z
f z dx
2 z
![Page 77: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/77.jpg)
Example 25X ~ U(0, 1), Y ~ U(0, 1) X Y
fX+Y = ?
Case 1: Case 2:
( , ) 1, 0 , 1f x y x y I(X), I(Y) 0
.Z X Y Define ( ) ?I Z (0,2)0
( ) ( , )z
Zf z f x z x dx 0< z 1 1 < z < 2
[0 ]I x z [ 1 1]I z x
0( )
z
Zf z dx z1
1( )Z z
f z dx
2 z
0 1( )
2 1 2Z
z zf z
z z
z
fZ(z)
![Page 78: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/78.jpg)
Example 26
x
y
f(x, y)
X ~ U(0, 1), Y ~ U(0, 1) X Y
(2 2) ?P X Y
0. ?5P X Y
( , ) 1, 0 , 1f x y x y
![Page 79: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/79.jpg)
Example 26
x
y
f(x, y)
X ~ U(0, 1), Y ~ U(0, 1) X Y
(2 2) ?P X Y
0. ?5P X Y
( , ) 1, 0 , 1f x y x y
x
y 1 1
0.5 2 2(2 2)
xP X Y dydx
1/ 4
2x + y = 2
![Page 80: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/80.jpg)
Example 26
x
y
f(x, y)
X ~ U(0, 1), Y ~ U(0, 1) X Y
(2 2) ?P X Y
0. ?5P X Y
( , ) 1, 0 , 1f x y x y
x
y 0.5P X Y 0.5 0.5P X Y
x y
= 0.5
x y
= 0.5
3/ 4
![Page 81: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/81.jpg)
Distributions of Multiplications and Quotients
Chapter 4-2Continuous Random Variables
![Page 82: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/82.jpg)
Distributions of Multiplications and and Quotients
Z XY /Z Y X( , )f x y 已知
) ?(Zf z ( , )f x y 已知
) ?(Zf z
1
|( ) ,
|Z
zf
x xz f x dx
( ) ,| |Zf z f x dxx zx
1
|( ) ,
|Z
zf
xxz f x dx
( ) ,| |Zf z f x dxzxx
1
|( ) ,
|Z
zf
xxz f x dx
( ) ,| |Zf z f x dxzxx
I(X), I(Y) 0 I(X), I(Y) 0
0,
1( )Z
z
xf x
xz f dx
0( ) ,Zf z f xx zx dx
X Yand X Yand
0(
1( ) )Z X Yf
z
xz x f x
xf d
0
( ) ( ) ( )Z X Y zxf dxz f x f x
![Page 83: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/83.jpg)
Example 27
Let X ~(1, ) and Y ~(2, ) be independent random variables. Find the pdf of Y/X.
Let X ~(1, ) and Y ~(2, ) be independent random variables. Find the pdf of Y/X.
X Y1 2~ ( ), ~ ( )X Y
( ?/ , )ZZ Y X f z
![Page 84: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/84.jpg)
Example 27
1 2 1 21 1 ( )
1 2
1( , ) , , 0
( ) ( )x yf x y x y e x y
0
( ) ,Zf z f xx zx dx
0
( ) ,Zf z f xx zx dx
1 2
1 2 1 (1 )
01 2
( ) ( )( ) ( )
x zZf z x zx e dx
1 2 2
1 2
11 (1 )
01 2( ) ( )
x zzx e dx
1 2 2
1 2 1 2
11 2
1 2
( )
( ) ( ) (1 )
z
z
2
1 2
11 2
1 2
( )
( ) ( ) (1 )
z
z
0z
Chapter 2Exercise
X Y1 2~ ( ), ~ ( )X Y
( ?/ , )ZZ Y X f z
![Page 85: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/85.jpg)
Example 27
1 2 1 21 1 ( )
1 2
1( , ) , , 0
( ) ( )x yf x y x y e x y
0
( ) ,Zf z f xx zx dx
0
( ) ,Zf z f xx zx dx
1 2
1 2 1 (1 )
01 2
( ) ( )( ) ( )
x zZf z x zx e dx
1 2 2
1 2
11 (1 )
01 2( ) ( )
x zzx e dx
1 2 2
1 2 1 2
11 2
1 2
( )
( ) ( ) (1 )
z
z
2
1 2
11 2
1 2
( )
( ) ( ) (1 )
z
z
0z
X Y1 2~ ( ), ~ ( )X Y
( ?/ , )ZZ Y X f z
![Page 86: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/86.jpg)
Conditional Densities
Chapter 4-2Continuous Random Variables
![Page 87: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/87.jpg)
Conditional Densities
Let X and Y be continuous random variables having j
pdf f. The conditional density fY|X is defined by
|
( , )( ) 0
( )( | )
0
XXY X
f x yf x
f xf y x
otherwise
![Page 88: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/88.jpg)
Facts | ( | )
( , )( ) 0
( )
0
X
X XY
f x yf x
f x
otherwise
f y x
|1. ( , ) (( | ) ).Y X Xf yf fxx y x
|2. ( )( | ) .Y X Yf yX fxY y
3. ( )Yf y ( , )f x y dx
| | ( )( )Y X Xf x dxf y x
|4. ( | ) ( | )Y XF y x P Y y X x ( , )
( )
P X x Y y
P X x
0
( , )lim
( )x
P x X x x Y y
P x X x x
0
( , )lim
( )
y x x
xx xx
Xx
f u v dudv
f u du
0
( , )lim
( )
y
xX
x f x v dv
f x x
( , )
( )
y
X
f x v dv
f x | ( | )Y X
yf v x dv
![Page 89: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/89.jpg)
Facts | ( | )
( , )( ) 0
( )
0
X
X XY
f x yf x
f x
otherwise
f y x
|1. ( , ) (( | ) ).Y X Xf yf fxx y x
|2. ( )( | ) .Y X Yf yX fxY y
3. ( )Yf y ( , )f x y dx
| | ( )( )Y X Xf x dxf y x
|4. ( | ) ( | )Y XF y x P Y y X x | ( | )Y X
yf v x dv
|5. ( | ) ( | )Y X
b
aP a Y b X x dvf y x
![Page 90: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/90.jpg)
Example 28
2( , ) 10 , 0 1.f x y xy x y
|
( )
( )
?
?( )
?
|
X
Y
Y X
f x
f y
f y x
( 0.5 | .25 ?0 )P Y X
![Page 91: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/91.jpg)
Example 28
2( , ) 10 , 0 1.f x y xy x y
|
( )
( )
?
?( )
?
|
X
Y
Y X
f x
f y
f y x
( 0.5 | .25 ?0 )P Y X
1 2( ) 10X xf x xy dy 2
0( ) 10
y
Yf y xy dx1
310
3 x
xy 2 2
05
yx y
310(1 ),
3x x 45 ,y0 1x 0 1y
2
| 3103
10( | )
(1 )Y X
xyf y x
x x
2
3
3,
1
y
x
0 1.x y
![Page 92: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/92.jpg)
Example 28
2( , ) 10 , 0 1.f x y xy x y
|
( )
( )
?
?( )
?
|
X
Y
Y X
f x
f y
f y x
( 0.5 | .25 ?0 )P Y X
2
| 3103
10( | )
(1 )Y X
xyf y x
x x
2
3
3,
1
y
x
0 1.x y
1
|0.5( 0.5 | 0.25) ( | 0.25)Y XP Y X f y dy
21
30.5
3
1 0.25
ydy
8 / 9
![Page 93: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/93.jpg)
Example 29
~ ( , ), | ~ ( )Y Exp
|
( )
( | )
?
?Y
Y
f y
f y
![Page 94: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/94.jpg)
Example 29
, |( , ) ( | ) ( )Y Yf y f y f
~ ( , ), | ~ ( )Y Exp
|
( )
( | )
?
?Y
Y
f y
f y
| ( | ) , 0yYf y e y
1
( ) , 0( )
ef
( )
,( )
ye
, 0y
,( ) ( , )Y Yf y f y d
1
( 1)
( ) ( )y
1
, 0( )
yy
( )
0( )ye d
![Page 95: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/95.jpg)
Example 29
, |( , ) ( | ) ( )Y Yf y f y f
~ ( , ), | ~ ( )Y Exp
|
( )
( | )
?
?Y
Y
f y
f y
( )
,( )
ye
, 0y
1( ) , 0
( )Yf y yy
,|
( , )( | )
( )Y
YY
f yf y
f y
( )
1
( )
( )
ye
y
1 ( )( )
( 1)
yy e
, 0y
![Page 96: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/96.jpg)
Example 29
~ ( , ), | ~ ( )Y Exp
|
( )
( | )
?
?Y
Y
f y
f y
,|
( , )( | )
( )Y
YY
f yf y
f y
( )
1
( )
( )
ye
y
1 ( )( )
( 1)
yy e
, 0y
| ~ ( 1, )Y y
![Page 97: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/97.jpg)
Multivariate Distributions
Chapter 4-2Continuous Random Variables
![Page 98: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/98.jpg)
Definitions
![Page 99: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/99.jpg)
Properties of Multivariate Distributions
11
1
( , , )1. ( , , )
nn
nn
F x xf x x
x x
1 12. ( , , ) 1n nf x x dx dx
1
11 1 1 1 13. ( , , ) ( , , )
n
n
b b
n n n n na aP a X b a X b f x x dx dx
1 1 1 14. ( ) ( , , , , )iX i i n i i nf x f x x x dx dx dx dx
![Page 100: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/100.jpg)
Definition Independence
Random variables X1, …, Xn are called independent if
11 1 1( , , ) ( ) ( ), , ,nn X X n nF x x F x F x x x
1 nX X
11 1 1( , , ) ( ) ( ), , ,nn X X n nf x x f x f x x x
![Page 101: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/101.jpg)
Example 30
( ) , 0i
i
xX i if x e x
1( )1 1( , , ) , 0 , ,nx xn
n nf x x e x x
![Page 102: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/102.jpg)
Example 311 nX X1 nX X ~ ( )iX Exp
1
1
min , , ( )
max , ,
?
?( )n Y
n Z
Y X X f y
Z X X f z
![Page 103: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/103.jpg)
Example 311 nX X1 nX X ~ ( )iX Exp
1
1
min , , ( )
max , ,
?
?( )n Y
n Z
Y X X f y
Z X X f z
( ) ( )YF y P Y y
( )( ) Y
Y
dF yf y
dy
1min , , nP X X y
11 min , , nP X X y
11 , , nP X y X y
11 nP X y P X y
( ) ( )
1 , 0iX i
x
F x P X x
e x
( ) ( )
1 , 0iX i
x
F x P X x
e x
1 , 0n ye y
, 0n yn e y
~ ( )Y Exp n
![Page 104: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/104.jpg)
Example 311 nX X1 nX X ~ ( )iX Exp
1
1
min , , ( )
max , ,
?
?( )n Y
n Z
Y X X f y
Z X X f z
( ) ( )ZF z P Z z
( )( ) Z
Z
dF zf z
dz
1max , , nP X X z
1 , , nP X z X z
1 nP X z P X z ( ) ( )
1 , 0iX i
x
F x P X x
e x
( ) ( )
1 , 0iX i
x
F x P X x
e x
1 , 0
nze z
11 , 0
nz zn e e z
![Page 105: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/105.jpg)
Important Theorem of Sums
To be proved in the next chapter.
![Page 106: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/106.jpg)
Important Theorem of Sums
:iX 有何意義
1 2 :nX X X 有何意義
![Page 107: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/107.jpg)
Important Theorem of Sums
1 2 :nX X X 有何意義
:iX 有何意義
![Page 108: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/108.jpg)
Important Theorem of Sums
2 1 1,
2 2
2 1
,2 2n
n
2 1,
2 2i
im
m 1
2 1 1,
2 2n
nm m
m m
![Page 109: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/109.jpg)
Important Theorem of Sums
熟記 !!! 靈活的將它們用於解題
![Page 110: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/110.jpg)
Multidimensional Changes of Variables
Chapter 4-2Continuous Random Variables
![Page 111: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/111.jpg)
Multidimensional Changes of Variables
Let X1, X2, …, Xn be continuous r.v.’s with jpdf 1 2, , , 1 2( , , , )
nX X X nf x x x
1 1 1
2 2 1
1
, ,
, ,
, ,
n
n
n n n
Y g X X
Y g X X
Y g X X
1 2, , , 1 2 ) ?( , , ,nY Y Y nf y y y
![Page 112: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/112.jpg)
Multidimensional Changes of Variables
Let X1, X2, …, Xn be continuous r.v.’s with jpdf 1 2, , , 1 2( , , , )
nX X X nf x x x
1 1 1
2 2 1
1
, ,
, ,
, ,
n
n
n n n
Y g X X
Y g X X
Y g X X
1 2, , , 1 2 ) ?( , , ,nY Y Y nf y y y
假設此函式為一對一
求反函式求反函式
1 1
2 12
1
1 , ,
, ,
, ,
n
n
n n n
X Y Y
X Y Y
h
h
Y YhX
![Page 113: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/113.jpg)
Multidimensional Changes of Variables
Let X1, X2, …, Xn be continuous r.v.’s with jpdf 1 2, , , 1 2( , , , )
nX X X nf x x x
1 2, , , 1 2 ) ?( , , ,nY Y Y nf y y y
1 1 1
2 2 1
1
, ,
, ,
, ,
n
n
n n n
Y g X X
Y g X X
Y g X X
一對一
求反函式求反函式1 1 1
1 2
2 2 2
1 2
1 2
n
n
n n n
n
x x xy y y
x x xy y y
x x xy y y
J
JacobinMatrix
JacobinMatrix
1 2 1 2, , , 1 2 , , , 1 2( , , , ) det( ,) ,( ),n nY Y Y nn X X Xf y y y f hJ h h
1 1
2 12
1
1 , ,
, ,
, ,
n
n
n n n
X Y Y
X Y Y
h
h
Y YhX
![Page 114: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/114.jpg)
Example 34
1 2, 1 2 1 2 1 2( , ) 4 , 0 , 1.X Xf x x x x x x
21 1
2 1 2
Y X
Y X X
1 2, 1 2( , ) ?Y Yf y y
![Page 115: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/115.jpg)
Example 34
1 2, 1 2 1 2 1 2( , ) 4 , 0 , 1.X Xf x x x x x x
21 1
2 1 2
Y X
Y X X
1 2, 1 2( , ) ?Y Yf y y
21 1
2 1 2
Y X
Y X X
求反函式求反函式1 1
2 2 1
X Y
X Y Y
1 1 1
1 2
2 2 2
1 2
1 2
n
n
n n n
n
x x xy y y
x x xy y y
x x xy y y
J
JacobinMatrix
JacobinMatrix
1/ 2112
3/ 2 1/ 211 2 12
0y
y yJ
y
![Page 116: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/116.jpg)
Example 34
1 2, 1 2 1 2 1 2( , ) 4 , 0 , 1.X Xf x x x x x x
21 1
2 1 2
Y X
Y X X
1 2, 1 2( , ) ?Y Yf y y
21 1
2 1 2
Y X
Y X X
求反函式求反函式1 1
2 2 1
X Y
X Y Y
JacobinMatrix
JacobinMatrix
1/ 2112
3/ 2 1/ 211 2 12
0y
y yJ
y
1 2 1 2, , , 1 2 , , , 1 2( , , , ) det( ,) ,( ),n nY Y Y nn X X Xf y y y f hJ h h
1
1det( )
2J
y
1
1det( )
2J
y
1 2 1 2, 1 2 , 1 2 11
1( , ) ,
2Y Y X Xf y y f y y yy
1 2 11
14
2y y y
y
![Page 117: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/117.jpg)
Example 34
1 2, 1 2 1 2 1 2( , ) 4 , 0 , 1.X Xf x x x x x x
21 1
2 1 2
Y X
Y X X
1 2, 1 2( , ) ?Y Yf y y
21 1
2 1 2
Y X
Y X X
求反函式求反函式1 1
2 2 1
X Y
X Y Y
JacobinMatrix
JacobinMatrix
1/ 2112
3/ 2 1/ 211 2 12
0y
y yJ
y
1
1det( )
2J
y
1
1det( )
2J
y
1 2 1 2, 1 2 , 1 2 11
1( , ) ,
2Y Y X Xf y y f y y yy
1 2 11
14
2y y y
y
2
1
2,
y
y 2 10 1y y
![Page 118: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/118.jpg)
Example 35
1 23 21 2 1 2( , ) 6 , , 0.x xf x x e x x
1 2Y X X ) ?(Yf y
1 1 10( ) ( , )
y
Yf y f x y x dx 1 13 2( )10
6y x y xe dx
1210
6y xye e dx 12
06
yxye e
26 1y ye e 2 36 , 0y ye e y
![Page 119: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/119.jpg)
Example 35
1 23 21 2 1 2( , ) 6 , , 0.x xf x x e x x
1 2Y X X ) ?(Yf y
11 1( ) ( )Y Yf y f y
Define 1 1 2
2 1
Y X X
Y X
1Y Y
1 2, 1 2 2( , )Y Yf y y dy
?
求反函式求反函式1 2
2 1 2
X Y
X Y Y
1 1 1
1 2
2 2 2
1 2
1 2
n
n
n n n
n
x x xy y y
x x xy y y
x x xy y y
J
JacobinMatrix
JacobinMatrix
0 1
1 1J
![Page 120: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/120.jpg)
Example 35
1 23 21 2 1 2( , ) 6 , , 0.x xf x x e x x
1 2Y X X ) ?(Yf y
Define 1 1 2
2 1
Y X X
Y X
1Y Y
1 2, 1 2 2( , )Y Yf y y dy
求反函式求反函式1 2
2 1 2
X Y
X Y Y
JacobinMatrix
JacobinMatrix
0 1
1 1J
det( ) 1J
det( ) 1J
1 2 1 2, , , 1 2 , , , 1 2( , , , ) det( ,) ,( ),n nY Y Y nn X X Xf y y y f hJ h h
1 2 1 2, 1 2 , 2 1 2( , ) 1 ( , )Y Y X Xf y y f y y y 2 1 23 2( )6 y y ye 1 22
2 16 , 0y ye y y
?11 1( ) ( )Y Yf y f y
![Page 121: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/121.jpg)
Example 35
1 23 21 2 1 2( , ) 6 , , 0.x xf x x e x x
1 2Y X X ) ?(Yf y
Define 1 1 2
2 1
Y X X
Y X
1Y Y
1 2, 1 2 2( , )Y Yf y y dy
求反函式求反函式1 2
2 1 2
X Y
X Y Y
JacobinMatrix
JacobinMatrix
0 1
1 1J
det( ) 1J
det( ) 1J
1 2 1 2, 1 2 , 2 1 2( , ) 1 ( , )Y Y X Xf y y f y y y 2 1 23 2( )6 y y ye 1 22
2 16 , 0y ye y y
11 22
206
y y ye dy
1 12 316 , 0y ye e y
11 1( ) ( )Y Yf y f y
![Page 122: Chapter 4-2 Continuous Random Variables 主講人 : 虞台文.](https://reader036.fdocuments.net/reader036/viewer/2022062301/56649f225503460f94c3aacf/html5/thumbnails/122.jpg)
Example 35
1 23 21 2 1 2( , ) 6 , , 0.x xf x x e x x
1 2Y X X ) ?(Yf y
Define 1 1 2
2 1
Y X X
Y X
1Y Y
1 2, 1 2 2( , )Y Yf y y dy
求反函式求反函式1 2
2 1 2
X Y
X Y Y
JacobinMatrix
JacobinMatrix
0 1
1 1J
det( ) 1J
det( ) 1J
1 2 1 2, 1 2 , 2 1 2( , ) 1 ( , )Y Y X Xf y y f y y y 2 1 23 2( )6 y y ye 1 22
2 16 , 0y ye y y
11 22
206
y y ye dy
1 12 316 , 0y ye e y
11 1( ) ( )Y Yf y f y
2 3( ) 6 , 0y yYf y e e y
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Example 36
此例非一對一,以上方法非直接可用,請參考講義。