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- 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.2. Let X and Y denote independent random variables with respective probability density func- tions fx(x) = 2x, 02. The continuous random variables X and Y have known joint probability density function f(x,y) given by £xy (x, y) = {(x + y) [(x+y)/8 0 ≤x≤ 2, 0≤y≤ 2 fxx otherwise Define the random variable z as Z = min(X,Y). a) Determine the probability density function fz(z) of the random variable z. b) Determine the probability Pr{0.5Suppose that X and Y are continuous random variables with joint pdf given by c(x²+y?) 0Let X and Y random variables have independent Gamma distributions with X-Gamma(1, 6) and Y-Gamma(2, B). a. Find the joint probability density of Z, = X + Y, Z, = X+Y a. Find the marginal pdf of Z2.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 (d) fZ|XY (z|x,y), (e) fX|YZ(x|y, z).Suppose X and Y are independent random variables. X iş uniformly distributed on (0,) and Y is exponentially distributed with 1=2. Find the joint density function f(x, y) of X and Y.Suppose that X and Y are independent and uniformly distributed random variables. Range for X is (−1, 1) and for Y is (0, 1). Define a new random variable U = XY, then find the probability density function of this new random variable.If the joint density function of X and Y is f(x, y) = 6e-2xe- -3y 0 < x <∞, 0 < y < ∞ and is equal to 0 outside this region, are the random variables independent?Suppose (X; Y ) is a continuous random vector with joint probability density function fx.y(x.v)= v Osxs2,0Suppose that the random variables Y1 and Y2 have joint probability density function, f(y1, y2), given 6y y2, 0syı < y2, y1 + y2 < 2 fV1, y2) = - 0, elsewhere Derive the marginal density of Y1. Derive the marginal density of Y2.(Hint: this will have to broken down into two parts!!!) Derive the conditional density of Y2 given Y1 = y1. Find P(Y2 < 1.1 | Y1 = .60). %D6. Let X and Y be continuous random variables with joint density function 24xy if 0 < x < 1,0 < y < 1 – x f (x, y) = 0. otherwise. Calculate E(Y|X = }).Recommended textbooks for youA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSONA First Course in Probability (10th Edition)ProbabilityISBN:9780134753119Author:Sheldon RossPublisher:PEARSON