Example_class_6_slides

Example_class_6_slides - Example class 6 STAT1301...

Info iconThis preview shows pages 1–8. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

This note was uploaded on 01/16/2012 for the course STAT 1301 taught by Professor Smslee during the Fall '08 term at HKU.

Page1 / 29

Example_class_6_slides - Example class 6 STAT1301...

This preview shows document pages 1 - 8. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online