Advanced Topics in Probability and Stochastic Processes
A solved assignment covering advanced concepts in probability and stochastic processes.
Ethan Wilson
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Advanced Topics in Probability and Stochastic Processes
1. Linear Transformations of Normal Random Variables are also Normal( )
( ) ( )
( ) ( ) ( )
( ) ( )
( )
( )
2
2
2
2
2
2
2
Let ~ , and ,
1 1
The p.d.f of , exp , , 0.
22
1 1
The c.d.f of , exp 22
Let
The c.d.f of ,
1 1
exp 22
X
x
X
Y
X N a b
X f x x x
X F x P X x t dt
Y aX b
Y F y P Y y
P aX b y
y b
P X a
t
−
= − − −
= = − −
= +
=
= +
−
=
= − −
( )
2
2 2
2 2
1 1
exp 22
Therefore, has also normal distribution with parameters , .
y b
a
t
dt
y a b dy
aa
Y a b a
−
−
−
= − − +
+
1. Linear Transformations of Normal Random Variables are also Normal( )
( ) ( )
( ) ( ) ( )
( ) ( )
( )
( )
2
2
2
2
2
2
2
Let ~ , and ,
1 1
The p.d.f of , exp , , 0.
22
1 1
The c.d.f of , exp 22
Let
The c.d.f of ,
1 1
exp 22
X
x
X
Y
X N a b
X f x x x
X F x P X x t dt
Y aX b
Y F y P Y y
P aX b y
y b
P X a
t
−
= − − −
= = − −
= +
=
= +
−
=
= − −
( )
2
2 2
2 2
1 1
exp 22
Therefore, has also normal distribution with parameters , .
y b
a
t
dt
y a b dy
aa
Y a b a
−
−
−
= − − +
+
( )
( ) ( )
( ) ( )
32 9 5
32 9 5 32 9 5 15 32 27 59
9 5 9 5 5 9
Therefore, the temperature in also follows normal distribution with mean 59 and SD 9.
o o
o o
o o
F C
F C
o
F C
F
= +
= + = + = + =
= = =2. Moment-generating function of the normal( )
( )
( )
( )
( )
2
2
22
2 2
22
2
Let ~ 0,1
1
The p.d.f of , exp 2 , .
2
The m.g.f of , exp
1
exp exp 2
2
1
exp 2 exp 2
2
Since exp exp 2 exp 2
exp 2 exp 2
exp 2
Z
Z
X N
Z f z z z
Z M t E tZ
tz z dz
t z t dz
tz z z tz
t z t
t
−
−
= − −
=
= −
= − − −
+ − = − +
= − + − −
= −
( )
21
Since exp 2 1
2 z t dz
−
− − =
( ) ( )
( ) ( )
32 9 5
32 9 5 32 9 5 15 32 27 59
9 5 9 5 5 9
Therefore, the temperature in also follows normal distribution with mean 59 and SD 9.
o o
o o
o o
F C
F C
o
F C
F
= +
= + = + = + =
= = =2. Moment-generating function of the normal( )
( )
( )
( )
( )
2
2
22
2 2
22
2
Let ~ 0,1
1
The p.d.f of , exp 2 , .
2
The m.g.f of , exp
1
exp exp 2
2
1
exp 2 exp 2
2
Since exp exp 2 exp 2
exp 2 exp 2
exp 2
Z
Z
X N
Z f z z z
Z M t E tZ
tz z dz
t z t dz
tz z z tz
t z t
t
−
−
= − −
=
= −
= − − −
+ − = − +
= − + − −
= −
( )
21
Since exp 2 1
2 z t dz
−
− − =
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Subject
Mathematics