4. Suppose two random variables X and Y has the following joint density function Cry, 22...
Let the random variables X, Y with joint probability density function (pdf) fxy(z, y) = cry, where 0 < y < z < 2. (a) Find the value of c that makes fx.y (a, y) a valid pdf. (b) Calculate the marginal density functions for X and Y (c) Find the conditional density function of Y X (d) Calculate E(X) and EYIX) (e Show whether X. Y are independent or not.
Suppose that X and Y are jointly continuous random variables with joint probability density function f(x,y) = {12rºy, 1 0, 0<x<a, 0<y<1 otherwise i) Determine the constant a ii) Find P(0<x<0.5, O Y<0.25) HE) Find the marginal PDFs fex) and y) iv) Find the expected value of X and Y. Le. E(X) and E(Y) v) Are X and Y independent? Justify your answer.
Let the joint density function of random variables X and Y be f(x,y) = 8 - x - y) for 0 < x < 2, 2 < y < 4 0 elsewhere Find : (1) P(X + Y <3) (11) P(Y<3 | X>1) (111) Var(Y | x = 1)
1. Consider two random variables X and Y with joint density function f(x, y)-(12xy(1-y) 0<x<1,0<p<1 otherwise 0 Find the probability density function for UXY2. (Choose a suitable dummy transformation V) 2. Suppose X and Y are two continuous random variables with joint density 0<x<I, 0 < y < 1 otherwise (a) Find the joint density of U X2 and V XY. Be sure to determine and sketch the support of (U.V). (b) Find the marginal density of U. (c) Find...
Please also explain 5. Following is the joint probability density function for the random variables X and Y. f(x,y) = U 2, 0 < x <1, 0 Sy < 1, 0 < x + y 51 elsewhere. 10, Find E(X +Y) and Var(X +Y).
Let X and Y be joint continuous random variables with joint density function f(x, y) = (e−y y 0 < x < y, 0 < y, ∞ 0 otherwise Compute E[X2 | Y = y]. 5. Let X and Y be joint continuous random variables with joint density function e, y 0 otwise Compute E(X2 | Y = y]
1. (10) Suppose the random variables X and Y have the joint probability density function 4x 2y f(x, y) for 0 x<3 and 0 < y < x +1 75 a) Determine the marginal probability density function of X. (6 pts) b) Determine the conditional probability of Y given X = 1. (4 pts)
# 6 If two random variables have the joint density f(x, y)=59 y?) for 0<x<1, 0<y<1 0 elsewhere a. Find the probability that 0.2 X<0.5 and 0.4<Y<0.6. b. Find the probability distribution function F(x, y). c. Are x and y independent?
Suppose X and Y are continuous random variables with joint density function 1 + xy 9 fx,y(2, y) = 4 [2] < 1, [y] < 1 otherwise 0, (1) (4 pts) Find the marginal density function for X and Y separately. (2) (2 pts) Are X and Y independent? Verify your answer. (3) (9 pts) Are X2 and Y2 independent? Verify your answer.
2. Suppose X and Y are continuous random variables with joint density function f(x, y) = 1x2 ye-xy for 1 < x < 2 and 0 < y < oo otherwise a. Calculate the (marginal) densities of X and Y. b. Calculate E[X] and E[Y]. c. Calculate Cov(X,Y).