Outline Outline Conditional Expected Value Conditional Expected - - PowerPoint PPT Presentation

SLIDE 1

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• G. Ahmadi

ME 529 - Stochastics

• G. Ahmadi

ME 529 - Stochastics

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

• G. Ahmadi

ME 529 - Stochastics

Conditional Distribution of Conditional Distribution of Y Y given event given event m m

( ) { } { } ( )

m P m y Y P m y Y P m y FY ∩ ≤ = ≤ = | | ( ) { } { } ( ) ( )

x F y x F x X P y Y x X P x X y F

X XY Y

, | = ≤ ≤ ∩ ≤ = ≤ ( ) ( ) ( ) ( ) ( )

∫ ∞

−

= ∂ ∂ = ≤

x XY X X XY Y

dx y x f x F x F y y x F x X y f

1 1,

1 , |

( ) ( ) ( ) ( )

1 2 2 1

2 1

, | x F x F dx y x f x X x y F

X X x x XY Y

− = ≤ <

∫

Similarly: Similarly: For m = {X For m = {X ≤ ≤ x}, x},

• G. Ahmadi

ME 529 - Stochastics

Noting , Noting , Conditional Distribution & Density given X = x Conditional Distribution & Density given X = x Similarly: Similarly:

( ) ( )

x x X x y F x X y F

Y x Y

∆ + ≤ < = =

→ ∆

| lim | ( ) ( ) ( ) ( ) ( ) ( ) ( )

dx x dF x y , x F x F x x F y , x F y , x x F lim x x | y F

X XY X X XY XY x Y

∂ ∂ = − + − + = =

→

∆ ∆

∆

( ) ( ) ( )

x f dy y x f x x y F

X y XY Y

∫ ∞

−

= =

1 1

, | ( ) ( ) ( ) ( ) x f y x f x x y f x y f

X XY Y Y

, | | = = =

( ) ( ) ( )

y f dx y x f y x F

Y x XY X

∫ ∞

−

=

1 1,

|

( ) ( ) ( )

y f y x f y x f

Y XY X

, | =

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• G. Ahmadi

ME 529 - Stochastics

Conditional Expected Value given X = x Conditional Expected Value given X = x Conditional expected value of a function of a Conditional expected value of a function of a random variable g(Y) random variable g(Y)

( ) { } ( ) ( )

∫

+∞ ∞ −

= dy m y f y g m Y g E

Y

| |

( ) { } ( ) ( )

∫

+∞ ∞ −

= = = dy x x y f y g x X Y g E

Y

| |

( ) { } ( ) ( ) ( )

∫

∞ + ∞ −

= = dy y x f y g x f x X Y g E

XY X

, 1 |

• G. Ahmadi

ME 529 - Stochastics

Integrating over y: Integrating over y: Noting Noting , ,

( ) ( ) ( )

y f y x f y x f

Y XY X

, | =

( ) ( ) ( )

z y f z y x f z y x f

Y XY X

| | , , | =

( ) ( ) ( )

z y f z y x f z y x f

Y X XY

| , | | , =

( ) ( ) ( )

∫

+∞ ∞ −

= dy z y f z y x f z x f

Y X X

| , | |

( ) ( )

y x f z y x f

X X

| , | =

( ) ( ) ( )

∫

+∞ ∞ −

= dy z y f y x f z x f

Y X X

| | |

For Markov processes: For Markov processes:

• r
• r
• G. Ahmadi

ME 529 - Stochastics

If X If Xi

i have the same mean & variance and form

have the same mean & variance and form a sequence of uncorrelated random variables: a sequence of uncorrelated random variables:

n X X X X

n

+ + + = ...

2 1

( ) ( )

n X X X X V

n 2 2 1

... − + + − =

{ } η

= X E n

X 2 2

σ σ =

{ }

2

1σ n n V E − =

• G. Ahmadi

ME 529 - Stochastics

When X When Xi

i are jointly normal with

are jointly normal with , density func , density functions tions

• f and become
• f and become

( ) ( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ + + + − =

2 2 2 2 2 1 2 1

2 ... exp 2 1 ,... σ σ π

n n n n

x x x x x f

∑

=

=

n j j

X n X

1

1

( )

∑

− =

j j

X X n V

2

1 ( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧− =

2 2 2

2 2 1 σ πσ x n exp n x f X

( )

( ) ( )

( )

v U e v n n v f

nv n n n V

2

2 2 3 1 2 1

2 1 2 1

σ

σ

− − − −

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − Γ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

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• G. Ahmadi

ME 529 - Stochastics

Density functions of Density functions of χ χ and and χ χ2

2 = Y:

= Y:

∑

=

=

n j j

X

1 2

χ

∑

=

= =

n j j

X Y

1 2 2

χ

( ) ( )

χ χ Γ σ χ

σ χ Χ

U e n f

n n n

2 2

2 1 2

2 2 2

− −

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ =

( )

( )

( )

y U e y n y f

y n n n

2 2

2 2 2 2

2 2 1

σ Χ

Γ σ

− −

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ =

• G. Ahmadi

ME 529 - Stochastics

If X If Xi

i have

have η η and and σ σ as mean and variance: as mean and variance:

n X X

i i

∑

=

( )

1

2 2

− − = ∑ n X X S

i i

{ } η

= X E n

X 2 2

σ σ =

• G. Ahmadi

ME 529 - Stochastics

Theorem Theorem: If is the mean of a random sample : If is the mean of a random sample

• f size n taken from a population having mean
• f size n taken from a population having mean

η η and variance and variance σ σ2

2 , then

, then is a random is a random variable whose distribution approaches a variable whose distribution approaches a standard normal distribution as n standard normal distribution as n → → ∞ ∞, i.e. , i.e. Note: Note:

X

n / X Z σ η − =

( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧− = 2 exp 2 1

2

z z f Z π

{ }

( )

z erf z Z P 2

2 =

≤

{ }

68 . 1 ≈ ≤ Z P

{ }

85 . 2 = ≤ Z P

{ }

997 . 3 = ≤ Z P

• G. Ahmadi

ME 529 - Stochastics

Size of a Sample for a Required Accuracy Size of a Sample for a Required Accuracy Example Example: Let z = 3, : Let z = 3, σ σ = 2, E = 0.01, then n = = 2, E = 0.01, then n = (9)(4)/10 (9)(4)/10-

• 4

4 data points are needed to estimate

data points are needed to estimate mean with probability 0.997 and error < 0.01 mean with probability 0.997 and error < 0.01 Let error and the set lead to Let error and the set lead to . The sample size needed is ; if . The sample size needed is ; if this is the sample size, then with probability of this is the sample size, then with probability of 2 2erf(z) erf(z) the error will not be more than the error will not be more than E E. .

η − = X E z Z ≤ n z E σ ≤

2 2 2

E z n σ =

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• G. Ahmadi

ME 529 - Stochastics

Stratonovich Stratonovich definition of Probability definition of Probability Density Function Density Function

( ) ( ) { }

x X E x f X − = δ

( ) { } ( ) ( ) ( )

x f dx x f x x x X E

X 1 1 X 1

= − δ = − δ

∫

+∞ ∞ −

( ) { } ( ) ( ) ( )

{ }

( ) ( ) { } ( ) ( )

∫ ∫ ∫

∞ + ∞ − ∞ + ∞ − +∞ ∞ −

= − = − = dx x f x g dx X x E x g dx x f X x x g E x g E

X X

δ δ

• G. Ahmadi

ME 529 - Stochastics

Transformation of Random Variable Transformation of Random Variable Using the New Definition, Y = Using the New Definition, Y = g(X g(X) )

( ) ( ) [ ] ( )

∫

+∞ ∞ −

− = dx x f y x g y f

X Y

δ

( ) [ ]

( ) ( )

∑

′ − = −

j j j

x g x x y x g δ δ

( )

( )

( )

( ) ( )

∑ ∫ ∑

′ = − =

∞ + ∞ − j j j X j X j Y

x g x f dx x f x x y f δ

• G. Ahmadi

ME 529 - Stochastics

• 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

• G. Ahmadi

ME 529 - Stochastics