Outline Outline Conditional Expected Value Conditional Expected - PowerPoint PPT Presentation

Outline Outline � Conditional Expected Value � Conditional Expected Value � Chapman � Chapman – – Kolmogorov Kolmogorov Equation Equation � � Sample Mean and Variance Sample Mean and Variance � Estimating Mean and Variance of � Estimating Mean and Variance of Random Data Random Data � Alternative Definition For � Alternative Definition For Probability Density Function Probability Density Function ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi Conditional Distribution of Y Y given event given event m m Conditional Distribution & Density given X = x Conditional Distribution of Conditional Distribution & Density given X = x { } ( ) ( ) ≤ ∩ P Y y m = = < ≤ + ∆ ( ) { } Noting , Noting , F y | X x lim F y | x X x x = ≤ = F Y y | m P Y y | m Y Y ( ) ∆ → x 0 P m ( ) ∂ ( ) F x , y y ∫ ∞ f x , y dy ( ) ( ) XY ≤ x}, + ∆ − ( ) For m = {X ≤ ∂ XY 1 1 ( ) F x x , y F x , y = = − x For m = {X x}, = = = F y | x x XY XY ( ) F y | x x lim ( ) ( ) ( ) Y Y + ∆ − f x ∆ → F x x F x dF x x 0 X X X X ( ) dx ∂ F x , y { } ( ) XY ( ) ≤ ∩ ≤ P X x Y y F x , y ∂ ( ) y 1 ( ) ( ) f x , y ( ) ( ) ≤ = = ∫ ∞ x = = = XY ≤ = = XY F y | X x f y | x f y | x x { } ( ) f y | X x f x 1 , y dx ( ) ( ) ( ) ≤ Y Y Y Y XY 1 f x P X x F x − F x F x X X X X ( ) ∫ ∞ x Similarly: ( ) ( ) Similarly: ∫ x f x 1 , y dx Similarly: Similarly: 2 f x , y f x , y dx ( ) ( ) = XY 1 = − XY ( ) XY F x | y f x | y ( ) ( ) < ≤ = x X F y | x X x 1 X ( ) ( ) f y f y Y 1 2 − Y F x F x Y X 2 X 1 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 1

( ) ( ) ( ) f x , y = f x , y | z Conditional expected value of a function of a Conditional expected value of a function of a f x | y XY ( ) Noting , Noting ( ) , = XY X f x | y , z f y ( ) X Y f y | z random variable g(Y) random variable g(Y) Y or or ( ) ( ) ( ) = f x , y | z f x | y , z f y | z +∞ { ( ) } ( ) ( ) ∫ XY X Y = E g Y | m g y f y | m dy Y − ∞ ( ) +∞ ( ) ( ) Integrating over y: ∫ Integrating over y: = f x | z f x | y , z f y | z dy { ( ) } +∞ ( ) ( ) ∫ X X Y = = = − ∞ E g Y | X x g y f y | x x dy Y − ∞ For Markov processes: For Markov processes: Conditional Expected Value given X = x Conditional Expected Value given X = x ( ) ( ) = ( ) +∞ ( ) ( ) ∫ f x | y , z f x | y = { ( ) } 1 + ∞ ( ) ( ) f x | z f x | y f y | z dy ∫ = = X X E g Y | X x g y f x , y dy ( ) X X Y − ∞ XY − ∞ f x X ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi ( ) ( ) + + + When X When X i i are jointly normal with are jointly normal with 2 2 X X ... X − + + − X X ... X X = 1 2 n X = 1 n V n ⎧ + + + ⎫ , density func , density functions tions n 2 2 2 x x ... x ( ) 1 = − f x ,... x exp ⎨ 1 2 n ⎬ 1 n ( ) n σ 2 2 ⎩ ⎭ π σ n 2 2 ( ) 1 n ∑ 1 ∑ 2 = If X i have the same mean & variance and form of and become = − If X i have the same mean & variance and form of and become X X V X X j j n n = j 1 j a sequence of uncorrelated random variables: a sequence of uncorrelated random variables: ⎧− ⎫ ( ) 2 − nv 1 n x n 3 ( ) 1 − ( ) ( ) = ⎨ ⎬ = f X x exp σ 2 f v v 2 e 2 U v σ { } η 2 ( ) − πσ 2 ⎩ 2 ⎭ V − n 1 2 n 1 ⎛ σ ⎞ ⎛ − ⎞ { } n 1 − ⎜ ⎟ Γ ⎜ ⎟ = σ n 1 σ 2 2 2 ⎜ ⎟ E X = n ⎝ ⎠ 2 ⎝ ⎠ 2 σ = n E V 2 n X n ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 2

( ) ∑ = ∑ 2 − X n ∑ X X n ∑ χ = = 2 2 i χ = Y X 2 i X j = j i X 2 i = S j 1 = j 1 − n n 1 χ and χ 2 2 = Y: Density functions of χ and χ Density functions of = Y: η and σ as mean and variance: have η and σ If X i If X i have as mean and variance: ( ) χ 2 − y { } η n 2 2 − 1 − ( ) ( ) ( ) ( ) χ = χ − χ = n 1 σ 2 σ 2 f e 2 U f y y 2 e 2 U y = σ Χ 2 Χ 2 n ⎛ ⎞ n ⎛ ⎞ E X n n σ Γ σ Γ σ = n ⎜ ⎟ n ⎜ ⎟ 2 2 2 2 2 ⎝ ⎠ ⎝ ⎠ 2 2 X n ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi Theorem : If is the mean of a random sample : If is the mean of a random sample Size of a Sample for a Required Accuracy Theorem X Size of a Sample for a Required Accuracy of size n taken from a population having mean of size n taken from a population having mean Z ≤ = X − η Let error and the set lead to Let error and the set lead to z − η E η and variance σ 2 2 , then X η and variance σ = σ , then is a random Z is a random σ 2 2 z σ ≤ / n = E z . The sample size needed is ; if . The sample size needed is ; if n n 2 E variable whose distribution approaches a variable whose distribution approaches a this is the sample size, then with probability of this is the sample size, then with probability of → ∞ ∞ , i.e. standard normal distribution as n → standard normal distribution as n , i.e. 2 2 erf(z) erf(z) the error will not be more than the error will not be more than E E . . { } ⎧− ⎫ 2 = 2 ( ) 1 z σ = 2, E = 0.01, then n = ( ) : Let z = 3, σ ≤ = ⎨ ⎬ P Z z 2 erf z Example : Let z = 3, Example = 2, E = 0.01, then n = f Z z exp π ⎩ 2 ⎭ 2 4 data points are needed to estimate (9)(4)/10 - -4 (9)(4)/10 data points are needed to estimate { } { } { } mean with probability 0.997 and error < 0.01 mean with probability 0.997 and error < 0.01 ≤ 1 ≈ ≤ 2 = ≤ 3 = Note: Note: P Z 0 . 68 P Z 0 . 85 P Z 0 . 997 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 3

Transformation of Random Variable Transformation of Random Variable Stratonovich Stratonovich definition of Probability definition of Probability Using the New Definition, Y = g(X g(X) ) Using the New Definition, Y = Density Function Density Function [ ] ( ) +∞ ( ) ( ) ∫ = δ − f y g x y f x dx Y X − ∞ ( ) ( ) { ( ) } = δ − { } δ − f X x E X x x x [ ( ) ] ∑ δ − = j { ( ) } +∞ ( ) ( ) ( ) ∫ g x y ( ) = δ − E g x E g x x X f x dx ′ g x X − ∞ j j + ∞ ( ) { ( ) } ∫ = δ − { ( ) } +∞ ( ) ( ) g x E x X dx ∫ δ − = δ − E X x x x f x dx − ∞ ( ) 1 X 1 1 − ∞ + ∞ ( ) ( ) ( ) f x ∫ ( ) + ∞ ( ) ( ) = ∫ ∑ ∑ g x f x dx = δ − = X j = f y x x f x dx f x ( ) X − ∞ Y j X ′ X − ∞ g x j j j ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi � Conditional Expected Value � Conditional Expected Value � Chapman � Chapman – – Kolmogorov Kolmogorov Equation Equation � Sample Mean and Variance � Sample Mean and Variance � Estimating Mean and Variance for � Estimating Mean and Variance for Random Data Random Data � Alternative Definition For � Alternative Definition For Probability Density Function Probability Density Function ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 4