Mathematical Statistics Quiz#2--2009

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

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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
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Unformatted text preview: à¬¶ áˆ» . (1) Find the pdf of Y àµŒ e Â± . (2) Find PáˆºY àµ’ yáˆ» . (3) Find EáˆºYáˆ» . 6. Let X à¬µ , X à¬¶ , Ú® , X à­¬ be n independent random variables with the same pdf PáˆºX àµŒ âˆ†xáˆ» àµŒ à¬µ à¬¶ àµ… Âµ à¬¶à®¢ à°® âˆ†x and PáˆºX àµŒ àµ†âˆ†xáˆ» àµŒ à¬µ à¬¶ àµ† Âµ à¬¶à®¢ à°® âˆ†x . Define the random variable W à­¬ àµŒ âˆ‘ X à­§ , à­¬ à­§à­€à¬µ n àµŒ 0,1,2, Ú®, W à¬´ àµŒ 0 , and assume that áˆºâˆ†xáˆ» à¬¶ àµŒ Ïƒ à¬¶ âˆ†t , therefore we have X à­²à¬¾âˆ†à­² àµŒ á‰Š X à­² àµ… Ïƒâˆšâˆ†t X à­² àµ† Ïƒâˆšâˆ†t . (1) Find EáˆºX à­§ áˆ» , VáˆºX à­§ áˆ» , EáˆºW à­¬ áˆ» , and VáˆºW à­¬ áˆ» . (2) Find the moment generating function(mgf) of X à­§ , denoted by M Â± áˆºtáˆ» . Express the mgf of W à­¬ , M Â² áˆºtáˆ» , in terms of M Â± áˆºtáˆ» . (3) Let âˆ†t àµŒ à­² à­¬ , show that W à­¬ converges in distribution to Báˆºtáˆ»~NáˆºÂµt, Ïƒ à¬¶ táˆ» . You can find lim âˆ†à­²Õœà¬´ M Â± áˆºtáˆ» àµŒ lim à­¬Õœà®¶ M Â± áˆºtáˆ» first then lim à­¬Õœà®¶ M Â² áˆºtáˆ» ....
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## 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.

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