Let X and Y be continuous random variables with joint p.d.f. f(x, y) = k(x + y), 0≤x≤y≤1 Find the (a) (b) value of the constant k joint c.d.f of X and Y F(x, y) marginal c.d.f. of X and marginal c.d.f. of Y. CX ginal ndf of Y
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- Let X and Y be continuous random variables with joint distribution function, F (x,y). Let g (X,Y) and h (X,Y) be functions of X and Y. PROVE Var(a + X) = Var (X)2. Let X and Y be continuous random variables with bivariate p.d.f. f(x, y) and let2) denőté marginal p.d.f. of Y. Show that E[X] = E(X\Y = y} f2(y)dy. %3D %3DI the variance of a poisson variate is 3. Find the probability that i) X = 0 i) ILet Y be a continuous random variable with √ ½ (2 f(y) = = Find the mean (u) and variance (o²) of Y. (2−y), 0≤ y ≤ 2 elsewhere.If X has an F distribution with ν1 and ν2 degrees offreedom, show that Y = 1Xhas an F distribution with ν2 and ν1 degrees of freedom.x is a uniform random variable in the interval (x,, x,). Find the expected value of x.Let X - U[0, 1]. Find the distribution of Y = 1 - X.Let Y1 and Y2 be independent random variables that are both uniformly distributed on the interval (0, 1). Find P( Y1 < 2Y2 | Y1 < 3Y2).Let Y1 < Y2 < Y3 be the order statistics of a random sample of size 3 froma distribution having the pdf f(x) = 2x, 0 < x < 1, zero elsewhere. Show thatZ1 = Y1/Y2, Z2 = Y2/Y3, and Z3 = Y3 are mutually independent.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 fx(x,y)= 1/x, 0<y<x<1 derive distribution of y and compute e(y) and var(y)Suppose that Y is a uniform continuous random variable on the interval (1,11). Calculate the expected value of the random variable (Y− 1)². Use one decimal place accuracy.