Outline Outline Conditional Distribution and Density Conditional - - PowerPoint PPT Presentation

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• G. Ahmadi

ME 529 - Stochastics

• G. Ahmadi

ME 529 - Stochastics

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

ME 529 - Stochastics

( ) ( ) { } ( ) { } { }

m P m x X P m x X P m x FX ∩ ≤ = ≤ = | | ξ

Conditional Distribution of X( Conditional Distribution of X(ξ ξ) given event m ) given event m Note that ((X( Note that ((X(ξ ξ) ) ≤ ≤ x) x) ∩ ∩ m) is the event m) is the event consisting of all outcomes consisting of all outcomes ξ ξ such that X( such that X(ξ ξ) ) ≤ ≤ x x and x and x ∈ ∈ m. The properties of F

• m. The properties of FX

X(x

(x | | m) are m) are similar to F similar to FX

X (x).

(x).

• G. Ahmadi

ME 529 - Stochastics

( ) ( ) { }

x m x x X x P dx m x dF m x f

x X X

∆ ∆ + ≤ ≤ = =

→ ∆

| lim | |

Conditional Density of X( Conditional Density of X(ξ ξ) given event m ) given event m ( )

| > m x f X

( )

1 | =

∫

+∞ ∞ −

dx m x f

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2

• G. Ahmadi

ME 529 - Stochastics

{ } ( )

> =< = ∫

+∞ ∞ −

X dx x xf X E

X

Expected Value Expected Value

( ) ( )

∑

− =

n n n X

x x P x f δ { }

n x x x X E

n

+ + + ≈ ...

2 1

For discrete random variable with For discrete random variable with Lebesgue Lebesgue Integral in Sample Space Integral in Sample Space

{ } ( ) ( ) { } ∫

∑ ∑ ∫

= ∆ + ≤ < = ∆ = =

+∞ −∞ = +∞ −∞ = ∞ + ∞ − S i i i i i i i i i

XdP x x X x P x x x f x dx x xf X E

• G. Ahmadi

ME 529 - Stochastics

( ) { } ( ) ( )

∫

+∞ ∞ −

= dx x f x g X g E

X

Expected Value of g(X) Expected Value of g(X)

( ) { } ( )

∑

=

i i i

x g P x g E

For discrete random variable For discrete random variable Expected value is a linear operator: Expected value is a linear operator:

( ) ( )

{ }

∑ ∑

= =

= ⎭ ⎬ ⎫ ⎩ ⎨ ⎧

n j j n j j

x g E X g E

1 1

• G. Ahmadi

ME 529 - Stochastics

{ }

2 2 2

η σ − = x E

Variance Variance { }

( )

∫

+∞ ∞ −

= = dx x f x x E m

X k k k

Moments Moments k kth th Central Moment Central Moment

( )

{ }

( ) ( )

∫

+∞ ∞ −

− = − = dx x f x x E

X k k k

η η µ

η =

1

m 1

0 =

m 1

0 =

µ

1 =

µ

2 2

σ µ =

3 2 3 3

2 3 η η µ + − = m m

( )

{ }

( )

∑

= −

− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = − =

k i i k i i k k

m i k x E 1 η η µ

Note: Note:

• G. Ahmadi

ME 529 - Stochastics

For a normal For a normal random variable random variable

( )

2 2

2

2 1

σ

σ π

x

e x f

−

=

{ }

( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ − ⋅ ⋅ ⋅ =

• dd

n even n n x E

n n

1 ... 3 1 σ

{ }

( )

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ + = − ⋅ ⋅ ⋅ =

+

1 2 ! 2 2 1 ... 3 1

1 2

k n k even n n X E

k k n n

σ π σ

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• G. Ahmadi

ME 529 - Stochastics

Tchevycheff Tchevycheff Inequality Inequality Proof Proof

( ) ( ) ( ) ( ) ( )

{ }

σ η σ σ η η σ

σ η σ η

k x P k dx x f k dx x f x dx x f x

k x k x

≥ − = ≥ − ≥ − =

∫ ∫ ∫

≥ − ≥ − +∞ ∞ − 2 2 2 2 2 2 2

{ }

2

1 k k X P ≤ ≥ − σ η

{ }

2

1 k k X P ≤ ≥ − σ η

• G. Ahmadi

ME 529 - Stochastics

Approximate Evaluation of Approximate Evaluation of Mean and Variance of g(X) Mean and Variance of g(X)

( ) { } ( ) ( ) ( ) ( ) 2

2

σ η η g g dx x f x g X g E ′ ′ + ≈ = ∫

∞ + ∞ −

( )

( )

2 2 2

σ η σ g

x g

′ ≈

{ }

X E = η

( )

{ }

2 2

η σ − = X E

• G. Ahmadi

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

ME 529 - Stochastics