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spr10_ex2sol - STA 4322/5328 Spr/2010 Major Exam1 PRINT...

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STA 4322/5328 – Spr/2010 Major Exam1 PRINT NAME _____________ For all problems, the sample mean, sample variance, and sample proportion are: ( 29 ) , ( ~ where : Proportion Sample 1 1 : variance Sample 1 : Mean Sample ^ 1 2 1 p n Bin Y n Y p Y Y n- Y n Y n i i n i i = - = = = Q.1 : Suppose Y 1,…, Y n denote a random sample from a population with an exponential distribution whose density is given by (where Y~Exp ( θ )): ( 29 2 / ) ( ) ( then ) ( ~ If : Note elsewhere 0 0 , 0 1 ) ( θ θ θ θ θ θ = = = - Y V Y E Exp Y y e y f y If Y (1) = min( Y 1,…, Y n ) then density function for Y (1) is: ( 29 ( 29 ( 29 = = - - elsewhere 0 0 , 0 * / 1 1 * / / 1 ) 1 ( 1 1 θ θ θ θ θ y e e n y g y ny P.1.a. Give the mean and variance of Y (1) ANSWER: ( 29 ( 29 ( 29 2 2 ) 1 ( ) 1 ( * * = = = = n Y V n Y E θ θ θ θ P.1.b. Give a function of Y (1) that is an unbiased estimate of θ : ( 29 ) 1 ( 1 ^ Y h = θ ANSWER: ) 1 ( 1 ^ nY = θ P.1.c. Give the variance of your estimator in Part P.1.b. ANSWER: ( 29 ( 29 2 2 2 2 ) 1 ( 2 ) 1 ( 1 ^ θ θ θ = = = = n n Y V n nY V V P.1.d. Give the variance of the unbiased estimator Y = 2 ^ θ ANSWER: n n Y V V 2 2 ^ ) ( θ θ = = Q.2: An engineer is interested in estimating the mean breaking strength of a new material to be used in satellites. Because the material is so expensive, she can only obtain n = 9 test measurements. She obtains a sample mean of 3000 psi and a sample standard deviation of 300 psi.
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