hw09 - ORIE 360/560 Fall 2007 Assignment 9 Due Thursday Nov...

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ORIE 360/560 Fall 2007 Assignment 9 Due Thursday, Nov. 15 at 2:00 pm Problem 1. Consider a casino in which gamblers are playing on identical slot machines. Assume each machine produces a jackpot on a particular pull of the level, independently from other pulls and other machines, with probability p . Suppose 20 gamblers are observed, and the numbers of times these gamblers must pull the lever before hitting their respective first jackpots (including the winning pull) are 22 , 24 , 13 , 15 , 22 , 1 , 11 , 11 , 31 , 35 , 27 , 13 , 25 , 6 , 11 , 21 , 41 , 3 , 12 , 10 . Find the MLE of p . Problem 2. Let X 1 , . . . , X n be a random sample from the Gamma( α, λ ) distribution, where the shape parameter α > 0 is known. Find the MLE for λ . Problem 3. Let X = ( X 1 , X 2 ) be a vector of two independent standard normal random variables. That is, X ∼ N 0 , 1 0 0 1 . Let A = a 11 a 12 a 21 a 22 be a 2 × 2 matrix of constants and b = ( b 1 , b 2 ) be a constant vector. The random vector Y = ( Y 1 , Y 2 ) = A X + b has the bivariate normal distribution, N ( μ, Σ). Find the mean vector μ and the covariance matrix Σ. Extra credit:
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