quiz02_2_solution

quiz02_2_solution - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Civil and Environmental Engineering 1.017 Computing and Data Analysis for Environmental Applications Quiz 2 Thursday, November 7, 2002 Please answer all questions on a separate piece(s) of paper with your name clearly identified: Problem 1 ( 60 points – you will receive partial credit for each part successfully completed) Consider a random variable x with the triangular PDF shown below: f x ( x ) (0,2/ a ) (0, 0) ( a , 0) x Note that this distribution has one unknown parameter a . Its mean and variance are: ] [ E x = 3 a ] [ Var x = 18 2 a You will receive 15 extra credit points if you prove that these expressions are correct . Extra Credit : E[x]= a 0 xf(x)dx= [(-2/a 2 )x 2 +2x/a]dx = a/3 Var[x] = a 0 (x-mu) 2 f(x)dx = (x-a/3) 2 (2/a-2x/a 2 )dx = a 2 /18 (work in between is required) Suppose that you obtain a random sample of 8 x values: [ x 1 , x 2 , …, x 8 ]. Perform the following steps: 1
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8 a) Devise an unbiased estimator a ˆ for the parameter a that depends only on the sample 1 8 mean m x = x i . i = 1 b) Derive the mean and variance of this estimator. Is the estimator consistent? Why? c)
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This note was uploaded on 11/29/2011 for the course CIVIL 1.00 taught by Professor Georgekocur during the Spring '05 term at MIT.

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quiz02_2_solution - MASSACHUSETTS INSTITUTE OF TECHNOLOGY...

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