Outline Outline Several Random Variables Several Random - - PowerPoint PPT Presentation

SLIDE 1

1

• G. Ahmadi

ME 529 - Stochastics

• G. Ahmadi

ME 529 - Stochastics

Outline Outline

• Several Random Variables

Several Random Variables

• Joint Distribution, Density Functions

Joint Distribution, Density Functions

• Independent Random Variables

Independent Random Variables

• Expected Value

Expected Value

• Covariance

Covariance

• Joint Characteristic Function

Joint Characteristic Function

• G. Ahmadi

ME 529 - Stochastics

Given a probability experiment Given a probability experiment ℑ ℑ: : (S, F, P), a random vector (S, F, P), a random vector X( X(ξ ξ) ) = ( = ( X X1

1(

(ξ ξ), X ), X2

2(

(ξ ξ), …, ), …, X Xn

n(

(ξ ξ) ) is ) ) is defined as a mapping of the defined as a mapping of the probability space unto a point of the probability space unto a point of the n n-

• dimensional Euclidean space

dimensional Euclidean space R Rn

n.

. That is X( That is X(ξ ξ) is defined by a certain ) is defined by a certain rule for every rule for every ξ ξ ∈ ∈ S. S.

• G. Ahmadi

ME 529 - Stochastics

Joint Distribution Function Joint Distribution Function Joint Density Function Joint Density Function

( ) { }

n n n

x X x X P x x F ≤ ≤ = ,..., ,...,

1 1 1 X

( ) ( )

n n n n

x x x x F x x f ∂ ∂ ∂ = ... ,..., ,...,

1 1 1 X X

Properties Properties

( ) ( )

1 ... ,..., ... ,..., ,

1 1

= = ∞ ∞ ∞

∫ ∫

+∞ ∞ − +∞ ∞ − n n

dx dx x x f F

X X

( ) ( ) { } ( )

∫ ∫

= ∈

D n n n

dx dx x x x f D X X P ... ,..., , ... ,...,

1 2 1 1

ξ ξ

SLIDE 2

2

• G. Ahmadi

ME 529 - Stochastics

The random variables X The random variables X1

1, X

, X2

2, …,

, …, X Xn

n are

are said to be independent if the events {X said to be independent if the events {X1

1

≤ ≤ x x1

1}, …, {

}, …, {X Xn

n ≤

≤ x xn

n} are independent for

} are independent for any x any x1

1, …,

, …, x xn

n.

.

( ) ( ) ( ) ( )

n n n

x F x F x F x x x F ... ,..., ,

2 2 1 1 2 1

=

X

( ) ( ) ( ) ( )

n n n x

x f x f x f x x x f ... ,..., ,

2 2 1 1 2 1

=

• G. Ahmadi

ME 529 - Stochastics

Expected Value Expected Value Covariance Covariance

( ) { } ( ) ( )

∫ ∫

+∞ ∞ − +∞ ∞ −

=

n n n n

dx ... dx x ,..., x f x ,..., x g ... X ,..., X g E

1 1 1 1 X

( )(

) { } { }

j i j i j j i i ij

X X E X X E c η η η η − = − − =

{ }

i i

X E = η

• G. Ahmadi

ME 529 - Stochastics

Characteristic and Density Function of Characteristic and Density Function of Fourier Transform Pair Fourier Transform Pair

( )

( )

{ } { }

X ω X ⋅ + +

= =

i X ... X i n

e E e E ,...,

n n

ω ω

ω ω Φ

1 1

1

( ) ( )

∫ ∫

+∞ ∞ − +∞ ∞ − ⋅

=

n i

dx ... dx f e ...

1

x ω

X x ω X

Φ

( ) ( ) ( )

∫ ∫

∞ + ∞ − ∞ + ∞ − ⋅ −

=

n i n

d ... d e ... f ω ω Φ π

1

2 1 ω x

x x ω X

• G. Ahmadi

ME 529 - Stochastics

If X If X1

1, X

, X2

2, …,

, …, X Xn

n are

are independent random variables independent random variables

( ) ( ) ( )

n n n

ω ω ω ω Φ Φ = Φ ... ,...,

1 1 1

SLIDE 3

3

• G. Ahmadi

ME 529 - Stochastics

Concluding Remarks Concluding Remarks

• Several Random Variables

Several Random Variables

• Joint Distribution, Density Functions

Joint Distribution, Density Functions

• Independent Random Variables

Independent Random Variables

• Expected Value

Expected Value

• Covariance

Covariance

• Joint Characteristic Function

Joint Characteristic Function

• G. Ahmadi

ME 529 - Stochastics