Chapter 4-2Continuous 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 4-2Continuous Random Variables
The Problem
( )Y g X( )Xf x 已知
( )Yf y =?
Example 11
2Y X( )Xf x 已知
( )Yf y =?
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
Example 17
( ) , 0xXf x e x
1/ , 0Y X
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
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
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
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
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
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
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
Jointly DistributedRandom Variables
Chapter 4-2Continuous Random Variables
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)
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)
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
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
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
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)
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)
Marginal Probability Density Functions
( ) ( , )
( ) ( , )
X
Y
f x f x y dy
f y f x y dx
Example 20
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
Example 20
1k
0
0.5
1
1.5
2
0
0.5
1
1.5
2
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
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
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
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
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
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
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
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
Example 20
, ( , ) , 0 , 1X Yf x y x y x y
, ) ?( ,X YF x y
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
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
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
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
Independence of Random Variables
Chapter 4-2Continuous Random Variables
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
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
Example 22
2 2( ) / 2( , )
,
x xy yf x y ce
x y
?
?
?X
Y
c
f
f
X Y ?
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
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
Example 23
1 2( )1 2( , ) , 0 ,x yf x y e x y
X Y ?
?( , )F x y
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
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
Distribution of Sums
Chapter 4-2Continuous Random Variables
The Problem
Given f(x, y) or F(x, y), and Z = (X, Y),
FZ(z) = ?
fZ(z) = ?
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
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
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
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
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
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
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
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 = ?
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
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
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
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)
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
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
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
Distributions of Multiplications and Quotients
Chapter 4-2Continuous Random Variables
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
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
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
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
Conditional Densities
Chapter 4-2Continuous Random Variables
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
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
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
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
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
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
Example 29
~ ( , ), | ~ ( )Y Exp
|
( )
( | )
?
?Y
Y
f y
f y
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
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
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
Multivariate Distributions
Chapter 4-2Continuous Random Variables
Definitions
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
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
Example 30
( ) , 0i
i
xX i if x e x
1( )1 1( , , ) , 0 , ,nx xn
n nf x x e x x
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
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
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
Important Theorem of Sums
To be proved in the next chapter.
Important Theorem of Sums
:iX 有何意義
1 2 :nX X X 有何意義
Important Theorem of Sums
1 2 :nX X X 有何意義
:iX 有何意義
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
Important Theorem of Sums
熟記 !!! 靈活的將它們用於解題
Multidimensional Changes of Variables
Chapter 4-2Continuous Random Variables
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
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
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
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
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
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
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
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
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
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
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
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
Example 36
此例非一對一,以上方法非直接可用,請參考講義。
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