Example_class_7_handout_solution

Example_class_7_handout_solution - THE UNIVERSITY OF HONG...

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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 . Solution. 1 E ( X ) = ˆ ∞ xf ( x ) dx = ˆ ∞ x · λ α Γ( α ) x α- 1 e- λx dx = λ α Γ( α ) ˆ ∞ x α e- λx dx = λ α Γ( α ) · Γ( α + 1) λ α +1 = α λ E ( X 2 ) = ˆ ∞ x 2 f ( x ) dx = ˆ ∞ λ α Γ( α ) x α +1 e- λx dx = λ α Γ( α ) ˆ ∞ x α +1 e- λx dx = λ α Γ( α ) · Γ( α + 2) λ α +2 = α ( α + 1) λ 2 V ar ( X ) = E ( X 2 )- E ( X ) 2 = α ( α + 1) λ 2- α 2 λ 2 = α λ 2 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 ∈ (-∞ , ∞ ) . (a) Show that ´ ∞-∞ e- x 2 / 2 dx = √ 2 π . Hence show that ´ ∞-∞ f ( x ) dx = 1. (b) Find the moment generating function of X. Hence show that the mean is equal to μ and the variance is equal to σ 2 . 2 (c) Let Z = X- μ σ . Show that Z is normal with mean 0 and variance 1. (d) Show that E ( Z k ) = 0 if k is an odd integer. (e) Find E ( | Z | ). (f) Find the mgf of Z 2 . What is the distribution of Z 2 ? Solution. (a) Note that ˆ ∞-∞ e- x 2 / 2 dx 2 = ˆ ∞-∞ ˆ ∞-∞ e- x 2 + y 2 2 dxdy Use the change of variables x = r cos θ , y = r sin θ . Then ˆ ∞-∞ ˆ ∞-∞ e- x 2 + y 2 2 dxdy = ˆ 2 π ˆ ∞...
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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_solution - THE UNIVERSITY OF HONG...

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