Find the density of Z = (X+ Y)2, where X and Y are independent uniform random variables over (-1, +1).
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- The density function of two random variables X and Y is ,-2(x+y) fx,r (x, y) =u(x)u(y)4e¯¾x*y) X,Y Find the mean value of the function e-*+),Let Xbe a continuous random variable with density f (x) = 24x-4 for x > 2. Then Var (X) is equal toX is a uniform random variable over the interval (3, 5). Find the density function of X for the interval (3, 5)
- Let X1 and X2 be two continuous random variableshaving the joint probability density f(x1, x2) = 4x1x2 for 0 < x1 < 1, 0 < x2 < 10 elsewhereFind the joint probability density of Y1 = X21 and Y2 = X1X2.Let (X,Y)' have density for r, y > 0, f (x, y) = (1+x)² (1+xy) > 0, otherwise. Show that X and X Y are independent, equidistriduted random variables and determine their distribution.Find the density of U = Y1 +Y2, where Y1 and Y2 are independent random variables with densities (yı – 1), 3 < Yı < 4, (y2 + 2), 0 < Y2 < 1, fy, (y1) : fy,(y2) = otherwise, otherwise.
- Let 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.Let X and Y be continuous random variables having a joint pdf given by f(x, y) = e-*, 0sysx 3).Random variables X and Y are independent and distributed as Uniform(−1,3).Find the density of random vector (Z,W), say f(z,w), where Z and W are defined asZ = X + 2Y, W = 2X −Y.
- Suppose that the random variables X, Y, Z have multivariate PDFfXYZ(x, y, z) = (x + y)e−z for 0 < x < 1, 0 < y < 1, and z > 0. Find (a) fXY(x, y), (b) fYZ(y, z), (c) fZ(z)Let X and Y be independent random variables with density f (x) = 3x² for 0 < x < 1. Then P (X+ Y < 1) is equal toFind the density of X = max(Y1,Y2), where Y1 and Y2 are independent random variables with densities 15 1/2 -1/2 fy, (y1) = { 21 0, (42 – 2), 0< y2 < 1, 0 < y1 < 1, fy, (Y2) = 4 otherwise, 0, otherwise.