Example_class_6_slides

# Example_class_6_slides - Example class 6 STAT1301...

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Unformatted text preview: Example class 6 STAT1301 Probability and Statistics I Chan Chi Ho, Chan Tsz Hin & Shi Yun October 25, 2011 Review:Some continuous distributions Gamma Function and Beta Function Γ( α ) = R ∞ e- x x α- 1 dx , α > Beta ( α , β ) = R 1 x α- 1 ( 1- x ) β- 1 dx = Γ( α + β ) Γ( α )Γ( β ) Uniform Distribution Let X ∼ U ( a , b ) , where b > a . The probability density function(pdf) of X is f ( x ) = ( 1 b- a if x ∈ [ a , b ] , otherwise Review: Some continuous distributions Exponential Distribution Let X ∼ Exp ( λ ) , where λ > . The PDF of X is f ( x ) = ( λ e- λ x if x = otherwise μ = 1 λ σ 2 = 1 λ 2 A exponential random variable can be used to discribe random time elapsing between unpredictable events. Review:Some continuous distributions Gamma Distribution Let X ∼ Γ( α , λ ) , where α , λ > . The pdf of X is f ( x ) = ( λ α x α- 1 e- λ x / Γ( α ) if x > otherwise μ = α λ σ 2 = α λ 2 Gamma distribution describes random time elapsing until the accumulation of a specific number of unpredictive events. Exponential distribution is a special case of Gamma distribution with α = 1. Chi squared distribution is a special case of Gamma distribution of Γ( . 5 , . 5 ) . Review:Some continuous distributions Chi-squared Distribution Let X ∼ χ 2 r , where r ( a positive integer) is the degree of freedom. The pdf of X is f ( x ) = ( x r / 2- 1 e- x / 2 / ( 2 r / 2 Γ( r / 2 )) if x > otherwise μ = r σ 2 = 2 r Review: Some continuous distributions I Normal Distribution Let X ∼ N ( μ , σ 2 ) , where μ ∈ (- ∞ , ∞ ) is the mean and σ 2 > 0 is the variance. The Pdf of X is: f ( x ) = 1 √ 2 πσ 2 e- ( x- μ ) 2 2 σ 2 , and- ∞ < x < ∞ Review: Some continuous distributions Some important points: (a) A normal distribution is symmetric about its mean. That is, if X ∼ N ( μ , σ 2 ) , then P ( X 6 μ- x ) = P ( X > μ + x ) In particular, Φ( x ) = 1- Φ( 1- x ) , where Φ( x ) is the cumulative distribution function....
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## This note was uploaded on 01/16/2012 for the course STAT 1301 taught by Professor Smslee during the Fall '08 term at HKU.

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Example_class_6_slides - Example class 6 STAT1301...

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