332:321Probability and Stochastic ProcessesExamination 2November 22, 2004You have 80 minutes to answer the following questions in the notebooks provided. You are permitted 1 double-sidedsheet of notes.Make sure that you have included your name, Rutgers netid and signature in each book used.(5points) Read each question carefully. All statements must be justified. Computations should be simplified as much aspossible. The following table may be helpful:x0.00.250.50.751.01.251.501.752.02.252.50(x)0.500.5990.6920.7730.8410.8940.9330.9600.9770.9880.9941.40 pointsSHORT ANSWER: Each part is a separate problem.(a)Yis a Gaussian(μ=2, σ=2)random variable. CalculateP[Y>3].(b)X1,X2andX3are independent identically distributed (iid) continuous uniform random variables. RandomvariableY=X1+X2+X3has expected valueE[Y]=0 and varianceσ2Y=9. What is the PDFfX1(x)ofX1?(c)Xis a Gaussian(μ=0, σ=1)random variable.Zis a Gaussian(0,4)random variable.XandZareindependent. LetY=X+Z. Find the correlation coefficientρofZandY. AreZandYindependent?
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