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e2q - ECE 514 Fall 2007 Exam 2 Due 12:30pm at ECE front...

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ECE 514 , Fall 2007 Exam 2: Due 12:30pm at ECE front office, October 25, 2007 Name: 75 mins.; Total 50 pts. 1. (14 pts.) Consider three real random variables X , Y , and Z . a. Suppose that Z = X 2 + Y 2 and that X and Y have a joint density that is uniform on the region { [ x, y ] : x, y [ - 1 , 1] } (i.e., the density function is constant in this region and is 0 elsewhere). Find the CDF of Z , F X ( z ) , for z 1 . b. Suppose X = Z 2 , Y = Z , and Z is uniformly distributed on [0 , 1] . Find the joint CDF of the pair ( X, Y ) , F X,Y ( x, y ) , x, y R . Are X and Y independent? 1
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2. (12 pts.) A computer in a lab has three kinds of users: students, professors, and intruders. If a student uses the computer, the amount of time spent logged in (in minutes) is a random variable with uniform [0 , 3] distribution. If a professor uses the computer, the time spent logged in has exp(1) distribution. If an intruder uses the computer, the time spent logged in is Rayleigh (1) . Suppose you are the system administrator and you notice that overnight someone logged in to the computer for 2 minutes. a. Use the maximum-likelihood rule to decide who used the computer overnight.
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