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1. Let X1, X 2 , Xn be independent and identically distributed random variables with n probability density and distribution functions f X (x) and Fx (x), respectively. In terms of f X (x) and Fx (x) (a) Find the pdf of the minimum, Y = min (X1, X 2' X1). (b) Find the pdf of the minimum, Y, when fx(x) = ae-axu(x). . 2. Let the random variable X have the gamma distribution with probability density function (pdf) given by 1 abxb-1 exp(-ax), x>0 Px (x) = T (b) 0, otherwise where a > 0, b>0. (a) Find the moment generating function of the distribution. (b) Find the mean and variance. (c) Find the cummulants, if they exist. 3. Under the hypothesis H0 and H1 the random variable X has the following conditional distributions: (a) Set up the Bayes test for equally likely hypotheses with cost function C00 = =0, Co1=1, C10 = 1, , C11 = 0 and determine the decision regions. (b) Find the average Bayes cost. 4. Let X1, X , X n be independent and identically distributed random variables with probability density that is parameterized by 0. For each of the following distributions, find the sufficient statistic for testing for the parameter Q (if one exists): 2 (a) A > 0, x 0 Ax: (b) fx(x)= 0+1 , X to X Assume to is known. [0, 5. Let the random variable X be an observation from an exponential distribution with known parameter 0, i.e., fx (x) = Ae-oxu(x). Consider the following hypothesis testing problem (where 01 > 80): H0: 0 = Oo H1: 0 = 01 (a) Design a Neyman-Pearson test. (b) Find the maximum attainable probability of detection as a function of the probability of false alarm.

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