Let X, Y be independent random variables with values in [0, 00) and the same PDF 2e-/. Let U = X² + Y*, V = Y/X. Compute the joint PDF fy.v and prove that U, V are independent.
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- Let X and Y be independent Exp(2) random variables. Define W = X + Y. a) Determine the correlation between X and W. The joint pdf of X and W is: fx,w(x, w) = 1^(2) e^(-1w) I{OLet x be random variable with E(x) = 2 and var(x) = 3. Verify that random variable x and the random variable y=-4x+8 are orthogonal. CS Scanned with CamScannerLet X and Y be two continuous random variables having joint pdffX,Y (x, y) = (1 + XY)/4, −1 ≤x ≤1, −1 ≤y ≤1.Show that X ^2 and Y ^2 are independent.Let X ~ Binom(16, 1/4 ) and Y~Geom( 1/2 ) be two independent random variables. Compute(i) Cov(2X, 3Y ),(ii) Cov(X, 4X).Let X be a continuous random variable with PDF 3 x > 1 x4 fx(x) = otherwise Find the mean and variance of x.Let X be a random variable with uniform distribution on the interval [-2,2]. Let Y be defined as Y = X5. Calculate the pdf of Y.Let the random variable X be defined on the support set (1,2) with pdf fX(x) = (4/15)x3, Find the variance of X.Let X and Y be independent random variables distributed as N(-1, 1) and N(1, 1), respectively. Find the mean and variance of U = 2X - 3Y. . Find the mean and variance of V = XY.Let X and Y be independent standard normal random variables. Determine the pdf of W = x² + y² using the moment generating function method. Find the mean and the variance of U if U = W.SEE MORE QUESTIONS
