( ) { ( ) } A random variable is subject to the following A - - PowerPoint PPT Presentation

SLIDE 1

1

• G. Ahmadi

ME 529 - Stochastics

• G. Ahmadi

ME 529 - Stochastics

Outline Outline

• Definition of a Random Variable

Definition of a Random Variable

• Probability Distribution Function

Probability Distribution Function

• Important Properties

Important Properties

• Probability Density Function

Probability Density Function

• Important Properties

Important Properties

• Common Density Functions

Common Density Functions

• G. Ahmadi

ME 529 - Stochastics

Given a random experiment Given a random experiment ℑ ℑ: (S, F, P), a real : (S, F, P), a real valued function X( valued function X(ξ ξ) defined on the ) defined on the probability space is called a random variable. probability space is called a random variable. A random variable is subject to the following A random variable is subject to the following requirement: requirement: 1.

• 1. For every real number x, the set

For every real number x, the set { {ξ ξ: X( : X(ξ ξ) ) ≤ ≤ x} is an event in F. x} is an event in F. 2.

• 2. P(x

P(x = = ∞ ∞) = 0, P(x = ) = 0, P(x = -

• ∞

∞) = 0. ) = 0.

• G. Ahmadi

ME 529 - Stochastics

( ) ( ) { }

x X P x FX ≤ = ξ

+∞ ≤ ≤ ∞ − x Probability Distribution Function of a Probability Distribution Function of a random variable random variable X( X(ξ ξ) is defines as ) is defines as

SLIDE 2

2

• G. Ahmadi

ME 529 - Stochastics

Important Properties Important Properties 1.

• 1. F(

F(-

• ∞

∞) = 0, F( + ) = 0, F( + ∞ ∞) = 1 ) = 1 2.

• 2. F(x) is non

F(x) is non-

• decreasing

decreasing 3.

• 3. F(x) is continuous from the right

F(x) is continuous from the right 4.

• 4. If F(x

If F(x0

0) = 0, F(x) = 0 for every x

) = 0, F(x) = 0 for every x ≤ ≤ x x0 5.

• 5. P{X(

P{X(ξ ξ) > x} = 1 ) > x} = 1 – – F(x) F(x) 6.

• 6. P{x

P{x1

1 < X

< X ≤ ≤ x x2

2} = F(x

} = F(x2

2)

) – – F(x F(x1

1)

) 7.

• 7. P{X = x} = F(x)

P{X = x} = F(x) – – F(x F(x -

• )

) 8.

• 8. P{x

P{x1

1 < X

< X ≤ ≤ x x2

2} = F(x

} = F(x2

2)

) – – F(x F(x1

1

• )

)

• G. Ahmadi

ME 529 - Stochastics

Density Function Density Function -

• Continuous X(

Continuous X(ξ ξ) )

( ) ( )

dx x dF x f

X X

=

Density Function Density Function -

• Discrete X(

Discrete X(ξ ξ) )

( ) ( )

∑

− =

i i i

x x P x f δ

{ }

i i

x x P P = =

• G. Ahmadi

ME 529 - Stochastics

Important Properties Important Properties 1. 1. 2. 2. 3. 3. 4. 4. 5.

• 5. Continuous

Continuous Random Random Variable Variable

( )

1 =

∫

+∞ ∞ −

dx x f

( ) ( )

∫ ∞

−

=

x

d f x F ξ ξ

{ } ( ) ( ) ( )

∫

= − = ≤ <

2 1

1 2 2 1 x x X X

dx x f x F x F x X x P

{ } ( ) x

x f x x X x P ∆ ≈ ∆ + ≤ <

( ) ( )

x x x X x P x f

x

∆ ∆ + ≤ < =

→ ∆

lim

( )

≥ x f

• G. Ahmadi

ME 529 - Stochastics

Normal Normal Laplace Laplace

( )

( )

2 2

2

2 1

σ η

σ π

− −

=

x x

e x f

( )

x

e x f

α

α

−

= 2

( )

σ η − + = x erf x Fx 2 1

f x η f x

SLIDE 3

3

• G. Ahmadi

ME 529 - Stochastics

Raleigh Raleigh Maxwell Maxwell ( ) ( )

x U e x x f

x

2 2

2 2 α

α

−

=

( )

2 2

2 2 3

2

α

α π

x

e x x f

−

=

( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ < ≥ = 1 x x x U

• G. Ahmadi

ME 529 - Stochastics

Cauchy Cauchy Weibull Weibull

( )

2 2

x / x f + = α π α

( )

⎪ ⎭ ⎪ ⎬ ⎫ ⎪ ⎩ ⎪ ⎨ ⎧ > =

− −

• therwise

x e kx x f

x x 1

β

α β

• G. Ahmadi

ME 529 - Stochastics

Beta Beta

( ) ( )

⎭ ⎬ ⎫ ⎩ ⎨ ⎧ ≤ ≤ − = elsewhere x x Ax x f

c b

1 1

( ) ( ) ( )

1 1 2 + Γ + Γ + + Γ = c b c b A

• G. Ahmadi

ME 529 - Stochastics

Poisson Poisson

( ) { }

! k a e k x P

k a −

= = ξ

( ) ( )

∑

∞ = −

− = !

k k a

k x k a e x f δ

f(x) x 4 2 8 6 F(x) x 4 2 8 6

SLIDE 4

4

• G. Ahmadi

ME 529 - Stochastics

Binomial Binomial

{ }

k n kq

p k n k x P

−

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = =

( ) ( )

∑

= −

− ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

n k k n k

k x q p k n x f δ 1 = + q p

f(x) x 4 2 8 6 F(x) x 4 2 8 6

• G. Ahmadi

ME 529 - Stochastics

Erlang Erlang Gamma Gamma

( ) ( ) ( )

x U e x b c x f

cx b b − +

+ Γ = 1

1

( ) ∫

∞ − −

= Γ

1

dy e y b

y b

( ) ( ) ( )

x U e x n c x f

cx n n − −

− =

1

! 1

( ) ( )

x U ce x f

cx −

=

Exponential Exponential

• G. Ahmadi

ME 529 - Stochastics

Concluding Remarks

Definition of a Random Variable Probability Distribution Function

• Important Properties

Probability Density Function

• Important Properties

Common Density Functions

Concluding Remarks Concluding Remarks

• Definition of a Random Variable

Definition of a Random Variable

• Probability Distribution Function

Probability Distribution Function

• Important Properties

Important Properties

• Probability Density Function

Probability Density Function

• Important Properties

Important Properties

• Common Density Functions

Common Density Functions

• G. Ahmadi

ME 529 - Stochastics