Outline Outline Several Random Variables Several Random - PowerPoint PPT Presentation

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 G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics Joint Distribution Function Joint Distribution Function Given a probability experiment ℑ ℑ : : Given a probability experiment (S, F, P), a random vector (S, F, P), a random vector ( ) { } = ≤ ≤ F x ,..., x P X x ,..., X x X 1 n 1 1 n n X( ξ ξ ) ) = ( = ( X ( ξ ξ ), X ( ξ ξ ), …, ( ξ ξ ) ) is X 1 ), X 2 ), …, X X n ) ) is X( 1 ( 2 ( n ( ( ) ∂ n F x x Joint Density Function ( ) ,..., Joint Density Function = defined as a mapping of the X defined as a mapping of the 1 n f x ,..., x X ∂ ∂ 1 n x ... x 1 n probability space unto a point of the probability space unto a point of the Properties Properties n - -dimensional Euclidean space dimensional Euclidean space R n . . R n n +∞ +∞ ( ) ( ) ∫ ∫ ∞ ∞ ∞ = = F , ,..., ... f x ,..., x dx ... dx 1 That is X( ξ ξ ) is defined by a certain X X ) is defined by a certain 1 n 1 n That is X( − ∞ − ∞ ξ ∈ ∈ S. rule for every ξ { ( ) ( ) } ( ) rule for every S. ∫ ∫ ξ ξ ∈ = P X ,..., X D ... f x , x ,..., x dx ... dx 1 n 1 2 n 1 n D G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics 1

The random variables X 1 , X 2 , …, X X n are The random variables X 1 , X 2 , …, n are Expected Value Expected Value said to be independent if the events {X said to be independent if the events {X 1 1 ≤ x ≤ x ≤ n ≤ x 1 }, …, {X X n x n } are independent for 1 }, …, { n } are independent for { ( ) } +∞ +∞ ( ) ( ) ∫ ∫ = E g X ,..., X ... g x ,..., x f x ,..., x dx ... dx X 1 n 1 n 1 n 1 n − ∞ − ∞ any x 1 any x 1 , …, , …, x x n n . . Covariance Covariance ( ) ( ) ( ) ( ) = F x , x ,..., x F x F x ... F x { ) ( ) } { } X ( 1 2 n 1 1 2 2 n n = − η − η = − η η c E X X E X X ij i i j j i j i j ( ) ( ) ( ) ( ) = { } f x , x ,..., x f x f x ... f x η = E X x 1 2 n 1 1 2 2 n n i i G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics { } { } ( ) ( ) ω + + ω Φ ω ω = = ⋅ ω X i X ... X i If X 1 , X 2 , …, X X n are If X 1 , X 2 , …, n are ,..., E e E e 1 1 n n X 1 n independent random variables independent random variables Characteristic and Density Function of Characteristic and Density Function of ( ) ( ) ( ) Φ ω ω = Φ ω Φ ω Fourier Transform Pair Fourier Transform Pair ,..., ... 1 n 1 1 n n ( ) +∞ +∞ ( ) ∫ ∫ Φ = ⋅ ω ω x x i ... e f dx ... dx X X 1 n − ∞ − ∞ ( ) + ∞ + ∞ ( ) 1 ∫ ∫ = − ⋅ Φ ω ω ω x x i ω f ... e d ... d ( ) X x π 1 n n − ∞ − ∞ 2 G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics 2

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 G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics 3