hw1 - 36-226 Summer 2010 Homework 1 Due June 30 1 Math...

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36-226 Summer 2010 Homework 1 Due June 30 1. Math review. (a) Simplify the following expression as much as possible. The x i are positive integers. There should be no Q signs in the result but there will be signs. ln n Y i =1 e a b x i x i ! ! . (1) (b) Find the value of x which maximizes the following function. Be sure to prove that it is a maximum. f ( x | n, a ) = a ln x + ( n - a ) ln(1 - x ) 0 < a < n (2) (c) Let ¯ X = 1 n n i =1 X i . Show that n X i =1 ( X i - μ ) 2 = n X i =1 ( X i - ¯ X ) 2 + n ( ¯ X - μ ) 2 . (3) Hint: Start with the left side. The first step is a trick. The rest is easy. Do not work from both sides. Proofs should be linear and sensical. 2. Probability review. Let X 1 and X 2 be independent, each with pdf f ( x ) = 1 β exp {- x/β } I [0 , ) ( x ) β > 0 . (4) (a) Find E [ X 1 ] and V ar [ X 1 ]. (b) Write down the joint density and integrate to find E [3 X 1 - 2 X 2 + 10]. Feel free to cite part (i) if necessary. (c) Use properties of expectation and the answer to (i) to verify your answer in (ii). (d) Find V ar
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