Outline Outline Conditional Distribution and Density Conditional - PowerPoint PPT Presentation

Outline Outline � Conditional Distribution and Density � Conditional Distribution and Density � Expected Value and Moments � Expected Value and Moments � Moments of Normal Random Variable � Moments of Normal Random Variable � Tchevycheff � Tchevycheff Inequality Inequality � � Approximate Evaluation of the Mean Approximate Evaluation of the Mean and Variance and Variance G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics ξ ) given event m ξ ) given event m Conditional Distribution of X( ξ Conditional Density of X( ξ ) given event m ) given event m Conditional Distribution of X( Conditional Density of X( ( ) { } { ( ) } ≤ ∩ ≤ ≤ + ∆ | | ( ) { ( ) } ( ) dF x m P x X x x m P X x m = = = ξ ≤ = | | | X lim F X x m P X x m f x m { } ∆ X ∆ → P m dx 0 x x ( ) ξ ) ≤ x) ∩ m) is the event > Note that ((X( ξ ) ≤ x) ∩ m) is the event | 0 Note that ((X( f X x m ξ such that X( ξ ) ≤ x consisting of all outcomes ξ such that X( ξ ) ≤ x consisting of all outcomes ∈ m. The properties of F and x ∈ m. The properties of F X (x | | m) are m) are and x X (x +∞ ( ) ∫ = | 1 f x m dx similar to F X (x). similar to F X (x). − ∞ G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics 1

= ∫ Expected Value Expected Value { } +∞ ( ) Expected Value of g(X) Expected Value of g(X) { ( ) } +∞ ( ) ( ) ∫ =< > = E X xf x dx X E g X g x f x dx X X − ∞ − ∞ For discrete random variable with For discrete random variable with For discrete random variable For discrete random variable ( ) ∑ ( ) { ( ) } ∑ ( ) + + + = δ − = { } ... x x x f x P x x E g x P g x ≈ 1 2 n E X X n n i i n n i Lebesgue Lebesgue Integral in Sample Space Integral in Sample Space Expected value is a linear operator: Expected value is a linear operator: +∞ +∞ ⎧ ⎫ } ∫ { } { } + ∞ ( ) ∑ ( ) ∑ { ∫ n ( ) n ( ) = = ∆ = < ≤ + ∆ = ∑ ∑ = E X xf x dx x f x x x P x X x x XdP ⎨ ⎬ E g X E g x i i i i i i i − ∞ j j S ⎩ ⎭ = −∞ = −∞ i i = = 1 1 j j G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics { } Variance Variance σ = − η 2 2 2 2 x E x 1 − ( ) For a normal For a normal = σ 2 2 f x e random variable random variable π σ 2 { } +∞ ( ) ∫ Moments Moments = = k k 0 = = η m E x x f x dx m 1 m k X 1 − ∞ ( ) ⎧ ⎫ { } ⋅ ⋅ ⋅ − σ { } n 1 3 ... 1 n n even +∞ th Central Moment Central Moment ( ) ( ) ( ) k th ∫ = k µ = − η = − η n ⎨ ⎬ k k E x x f x dx E x k X − ∞ ⎩ ⎭ 0 n odd µ 0 = µ 1 = µ = σ µ = − η + η 3 1 2 0 m 3 m 2 ( ) 2 3 3 2 ⎧ ⋅ ⋅ ⋅ − σ ⎫ n { } 1 3 ... n 1 n even ⎪ ⎪ n = ⎨ ⎬ E X 2 { } ⎛ ⎞ σ + = + k 2 k 1 Note: Note: k k 2 ! 2 1 ( ) ∑ ( ) k n k ⎪ ⎪ ⎜ ⎟ µ = − η k = − 1 η i i π E x ⎜ ⎟ m ⎩ ⎭ − k k i ⎝ i ⎠ = i 0 G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics 2

{ } Tchevycheff Tchevycheff Inequality Inequality 1 Approximate Evaluation of Approximate Evaluation of − η ≥ σ ≤ P X k Mean and Variance of g(X) 2 Mean and Variance of g(X) k σ = ∫ 2 { ( ) } + ∞ ( ) ( ) ( ) ( ) 2 ′ ′ ≈ η + η E g X g x f x dx g g +∞ ( ) ( ) ( ) ( ) Proof Proof ∫ ∫ σ = − η 2 ≥ − η 2 − ∞ 2 x f x dx x f x dx − ∞ − η ≥ σ x k { } ( ) ∫ ( ) ≥ σ = σ − η ≥ σ ′ 2 2 2 2 σ ≈ η σ k f x dx k P x k 2 2 2 g ( ) − η ≥ σ x k g x { } η = E X { } 1 − η ≥ σ ≤ P X k { } 2 k ( ) σ = − η 2 2 E X G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics Concluding Remarks Concluding Remarks � Conditional Density and Distribution � Conditional Density and Distribution � Expected Value and Moments � Expected Value and Moments � Moments of Normal Random Variable � Moments of Normal Random Variable � Tchevycheff � Tchevycheff Inequality Inequality � Approximate Evaluation of the Mean � Approximate Evaluation of the Mean and Variance and Variance G. Ahmadi G. Ahmadi ME 529 - Stochastics ME 529 - Stochastics 3