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

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 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi ℑ : (S, F, P), a real Given a random experiment ℑ Probability Distribution Function of a Given a random experiment : (S, F, P), a real Probability Distribution Function of a ξ ) defined on the ξ ) is defines as valued function X( ξ X( ξ valued function X( ) defined on the random variable X( random variable ) is defines as 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 = ξ ≤ − ∞ ≤ ≤ +∞ F X x P X x x requirement: requirement: 1. For every real number x, the set For every real number x, the set 1. ξ : X( ξ ) ≤ x} is an event in F. { ξ : X( ξ ) ≤ { x} is an event in F. ∞ ) = 0, P(x = - ∞ ∞ ) = 0. = ∞ 2. P(x P(x = ) = 0, P(x = - ) = 0. 2. ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 1

ξ ) Continuous X( ξ Important Properties Important Properties Density Function Density Function - - Continuous X( ) - ∞ ∞ ) = 0, F( + ∞ ) = 1 ) = 0, F( + ∞ 1. F( 1. F(- ) = 1 ( ) ( ) dF x 2. F(x) is non F(x) is non- -decreasing decreasing 2. = X f x X dx 3. 3. F(x) is continuous from the right F(x) is continuous from the right ) = 0, F(x) = 0 for every x ≤ ≤ x 4. If F(x If F(x 0 x 0 4. 0 ) = 0, F(x) = 0 for every x 0 Discrete X( ξ ξ ) Density Function - - Discrete X( ) ξ ) > x} = 1 Density Function P{X( ξ 5. P{X( 5. ) > x} = 1 – – F(x) F(x) < X ≤ ≤ x 6. P{x P{x 1 x 2 } = F(x 2 ) – – F(x F(x 1 ) 6. 1 < X 2 } = F(x 2 ) 1 ) ( ) ( ) ∑ = δ − { } f x P x x = = - ) 7. P{X = x} = F(x) P{X = x} = F(x) – – F(x F(x - ) 7. P P x x i i i i i ≤ x < X ≤ - ) - 8. P{x 8. P{x 1 1 < X x 2 2 } = F(x } = F(x 2 2 ) ) – – F(x F(x 1 ) 1 ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi f Normal Normal Important Properties Important Properties ( ) ≥ 1. 1. f x 0 ( ) − η 2 x − ( ) 1 +∞ ( ) = ∫ σ 2 = f x e 2 f x dx 1 2. 2. x π σ 2 − ∞ ( ) ( ) ∫ ∞ x = ξ ξ F x f d 3. 3. − η η − ( ) 1 x x = + F x x erf { } ( ) ( ) ( ) σ ∫ x 2 < ≤ = − = 2 4. 4. P x X x F x F x f x dx f 1 2 X 2 X 1 x 1 { } ( ) x < ≤ + ∆ ≈ ∆ 5. Continuous 5. Continuous P x X x x f x α ( ) Laplace Laplace = 2 − α ( ) x Random Random f x e < ≤ + ∆ ( ) P x X x x = f x lim Variable Variable ∆ ∆ → x x 0 x ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 2

Raleigh 2 Raleigh x − ( ) x ( ) Cauchy Cauchy = α 2 f x e 2 U x α 2 α π ( ) / = α ≥ ⎧ ⎫ f x ( ) 1 x 0 = ⎨ ⎬ + U x 2 2 < x ⎩ 0 x 0 ⎭ Maxwell Maxwell Weibull Weibull ⎧ ⎫ β 2 ⎪ β − − α > ⎪ x 1 x − ( ) kx e x 0 ( ) 2 = = ⎨ ⎬ α f x 2 2 f x x e 2 x ⎪ ⎪ π α ⎩ ⎭ 3 0 otherwise ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi ( ) ( ) ⎧ ⎫ Γ + + − ≤ ≤ b c Poisson b c 2 Poisson Beta ( ) Ax 1 x 0 x 1 Beta = = ⎨ ⎬ f x A ( ) ( ) Γ + Γ + f(x) b 1 c 1 ⎩ 0 elsewhere ⎭ k { ( ) } a − ξ = = a P x k e k ! 2 4 6 8 x ∞ k ( ) a ( ) ∑ F(x) = − δ − a f x e x k k ! = k 0 2 4 6 8 x ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 3

n Binomial Binomial Erlang Erlang ( ) c ( ) = − − n 1 cx f x x e U x ( ) − f(x) n 1 ! ⎛ ⎞ { } n − = = ⎜ ⎟ k q n k P x k ⎜ ⎟ p + b 1 Gamma Gamma ( ) c ( ) ⎝ ⎠ k = − b cx f x x e U x ( ) Γ + 2 4 6 8 b 1 x ⎛ ⎞ F(x) n n ( ) ∑ ( ) = ⎜ ⎟ − δ − k n k ( ) ∫ f x ⎜ ⎟ p q x k ∞ Γ = − − ⎝ ⎠ b 1 y k b y e dy = k 0 0 + q = ( ) ( ) p 1 = − cx Exponential f x ce U x Exponential 2 4 6 8 x ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi Concluding Remarks Concluding Remarks Concluding Remarks � Definition of a Random Variable � Definition of a Random Variable � Definition of a Random Variable � � Probability Distribution Function � Probability Distribution Function Probability Distribution Function - Important Properties Important Properties - - Important Properties � � Probability Density Function � Probability Density Function Probability Density Function - Important Properties Important Properties - - Important Properties � Common Density Functions � � Common Density Functions Common Density Functions ME 529 - Stochastics G. Ahmadi ME 529 - Stochastics G. Ahmadi 4