Example_class_7_handout

Example_class_7_handout - THE UNIVERSITY OF HONG KONG...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 7 Review Quantiles of a distribution The β quantile of a probability distribution function F X of a random variable X is defined to be F- 1 X ( β ) = inf { x ∈ R : F X ( x ) ≥ β } . Transformation of pdf Let X be a continuous random variable distributed on a space S with pdf f x ( x ). Let Y = g ( X ) where g is a function such that g- 1 exists. Then the pdf of Y can be obtained by f Y ( y ) = f X ( g- 1 ( y )) d dy g- 1 ( y ) , y ∈ g ( S ) . Problems Problem 1. Let X ∼ Γ( α,λ ) , where α,λ > . The pdf of X is f ( x ) = λ α Γ( α ) x α- 1 e- λx if x > , if x ≤ . Find the mean and variance of X . 1 Problem 2. The random variable X is said to have a normal distribution with mean μ and variance σ 2 (Gaussian distribution) if its pdf is defined by f ( x ) = 1 √ 2 πσ e- ( x- μ ) 2 2 σ 2 , x ∈ (-∞ , ∞ ) ....
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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_7_handout - THE UNIVERSITY OF HONG KONG...

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