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Unformatted text preview: 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 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 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) ....
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 Spring '08
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 Variance, probability density function, Cumulative distribution function, real random variables, bivariate Gaussian density

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