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please show all steps with justifications 21. Assume the joint density function of X and Y...
Is a joint density function? If yes, assume it is the joint density function of r.v.s X and Y , and compute the marginal densities of X and Y . f(r,y) = { " 0 <y<<11 , otherwise
Let X, Y be jointly continuous with joint density function (pdf) fx,y(x, y) *(1+xy) 05 x <1,0 <2 0 otherwise (a) Find the marginal density functions (pdf) fx and fy. (b) Are X and Y independent? Why or why not?
7. Suppose that the joint density of X and Y is given by f(x,y) = e-ney, if 0 < x < f(z, y) = otherwise. Find P(X > 1|Y = y)
(pts) 1. The joint probability density of X and Y is given by . 0<x<1 and 0 <y<2 otherwise d) Find Cov(X,Y). a) Verify that this is a joint probability density function. b) Find P(x >Y). ) Find Pſy>*<51 c) Find the correlation coefficient of X and Y (Pxy).
8), Let X and Y be continuous random variables with joint density function f(x,y)-4xy for 0 < x < y < 1 Otherwise What is the joint density of U and V Y
7. The joint pdf of two random variables X and Y is given by 0sxs3,0s y<5 fx(x,y) 15' 0, otherwise Find Cov(X,y)
Q. Suppose the joint probability density function of X and Y is (a) Show that the value of constant ?=12/11 (b) Find the marginal density function of X, i.e., fX(x). (c) Find the conditional probability density of X given Y = y, i.e., fX|Y(x|y). fxy(x, y) = s k(2 - x + y)x 1 0 0 < x < 1,0 = y = 1 otherwise
Random variables X and Y have the following joint probability density function, fx,y(x, y) = {c)[4] < 15.36, 1y| < 15.367 1.36} 0, 0.w. where cis a constant. Calculate P(Y – X| < 8.41).
Problem 5. The joint density of X and Y is given by e" (z+y) fx.-otherwise. İf 0 < x < oo, 0 < y < 00, Consider the random variable Z-; a) Find the cumulative distribution function of Z b) What is the probability density function of Z?
(8pts) 1. The joint probability density of X and Y is given by + 0<x<1 and 0 <y< 2 otherwise a) Verify that this is a joint probability density function. b) Find P(x >Y). o) Find Pſy > for< d) Find Cov(X,Y). e) Find the correlation coefficient of X and Y (Pxy).