Mathematical Statistics Quiz#2--2009

Mathematical Statistics Quiz#2--2009 - (1 Find the pdf...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Mathematical Statistics Quiz #2 5/21/2009 1. Let X and Y be random variables for which the joint pdf is as follows: fሺx, yሻൌ൜ 2ሺx൅yሻ, 0 ൏ x ൏ ݕ ൏ 1, 0, elsewhere . Find the pdf of ZൌX൅Y by (1) Convolution, (2) Distribution function, (3) Transformation technique by letting WൌX . Draw an appropriate picture for each method. 2. (1) Let X ,X ,ڮ,X be independent Poisson random variables with X ~ Pሺλ ሻ, kൌ1 ,ڮ,n . Find the distribution of S ൌ ∑ X ୧ୀଵ . (2) If X ~ Pሺµሻ and the conditional distribution of Y , given Xൌx , is Bሺx, pሻ . Find the distribution of Y . 3. Let X ~ Gammaሺα, βሻ . (1) Find the moment generating function (mgf) of aX ൅ b . (2) Use (1) to find the mean and distribution of 5X . (3) Let Y ~ Gammaሺγ, βሻ be independent of X . Find the mgf of 2X െ 3Y ൅ 1 . 4. Let X ,ڮ,X be a random sample from exponential distribution with pdf ݂ሺݔሻൌߣ݁ ିఒ௫ , show that the distribution of 2λ ∑ X ୧ୀଵ is independent of λ . Identify the distribution. 5. Let
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ଶ ሻ . (1) Find the pdf of Y ൌ e ± . (2) Find PሺY ൒ yሻ . (3) Find EሺYሻ . 6. Let X ଵ , X ଶ , ڮ , X ୬ be n independent random variables with the same pdf PሺX ൌ ∆xሻ ൌ ଵ ଶ ൅ µ ଶ஢ మ ∆x and PሺX ൌ െ∆xሻ ൌ ଵ ଶ െ µ ଶ஢ మ ∆x . Define the random variable W ୬ ൌ ∑ X ୧ , ୬ ୧ୀଵ n ൌ 0,1,2, ڮ, W ଴ ൌ 0 , and assume that ሺ∆xሻ ଶ ൌ σ ଶ ∆t , therefore we have X ୲ା∆୲ ൌ ቊ X ୲ ൅ σ√∆t X ୲ െ σ√∆t . (1) Find EሺX ୧ ሻ , VሺX ୧ ሻ , EሺW ୬ ሻ , and VሺW ୬ ሻ . (2) Find the moment generating function(mgf) of X ୧ , denoted by M ± ሺtሻ . Express the mgf of W ୬ , M ² ሺtሻ , in terms of M ± ሺtሻ . (3) Let ∆t ൌ ୲ ୬ , show that W ୬ converges in distribution to Bሺtሻ~Nሺµt, σ ଶ tሻ . You can find lim ∆୲՜଴ M ± ሺtሻ ൌ lim ୬՜ஶ M ± ሺtሻ first then lim ୬՜ஶ M ² ሺtሻ ....
View Full Document

This note was uploaded on 05/09/2011 for the course DIF 1486 taught by Professor Yow-jenjou during the Spring '11 term at National Chiao Tung University.

Ask a homework question - tutors are online