X = √-2 log U cos(2πV), Find the joint distribution of X and Y. (Hint: you may want to first compute the distribution of √-2 log U). Y:= √-2 log U sin(27V)
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- B A random process is defined as X (t) A. cos (100 t + 0) where 'A'is a normal random variable with zero mean and unity variance. O is a uniformn random variable over ( n, T) and is independent of A. F'ind the auto correlation function of X (t).Let X1, X2,... , Xn be independent Exp(A) random variables. Let Y = X(1)min{X1, X2, ... , Xn}. Show that Y follows Exp(nA) dis- tribution. Hint: Find the pdf of YLet random variables X and Y have the joint pdf fX,Y (x, y) = 4xy, 0 < x < 1, 0 < y < 1 0, otherwise Find the joint pdf of U = X^2 and V = XY.
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- Assume that X and Y are independent random variables where X has a pdf given by fx(x) = 2aI(0,1)(x) and Y has a pdf given by fy(y) = 2(1– y)I(0,1)(y). Find the distribution of X + Y.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 X and Y be a random variables having pdf f(x,y)=2xy 0<x<y<1 Find P(X/Y<1/2)