2. Let X and Y be two random variables with a joint distribution (discrete or continuous)....
Problem 2 Suppose two continuous random variables (X, Y) ~ f(x,y). (1) Prove E(X +Y) = E(X)+ E(Y). (2) Prove Var(X + Y) = Var(X) + Var(Y)2Cov(X, Y). (3) Prove Cov(X, Y) E(XY)- E(X)E(Y). (4) Prove that if X and Y are independent, i.e., f(x, y) Cov(X, Y) 0. Is the reverse true? (5) Prove Cov (aX b,cY + d) = acCov(X, Y). (6) Prove Cov(X, X) = Var(X) fx (x)fy(y) for any (x,y), then =
4. Let X and Y be continuous random variables with joint density function f(x, y) = { 4x for 0 <x<ys1 otherwise (a) Find the marginal density functions of X and Y, g(x) and h(y), respectively. (b) What are E[X], E[Y], and E[XY]? Find the value of Cov[X, Y]
Let X and Y be continuous random variables with following joint pdf f(x, y): y 0<1 and 0<y< 1 0 otherwise f(x,y) = Using the distribution method, find the pdf of Z = XY.
Problem 47.18 Let X and Y be discrete random variables with joint distribution defined by the following table Y X 2 345 Py(y) 0.05 0.05 0.15 0.05 0.30 0.40 0 0.05 0.15 0.10 0 0.40 0.30 2 px(x 0.50 0.20 0.25 0.05 1 For this joint distribution, E(X) = 285, E(Y) = 1 . Calculate Coy(X,Y)
2. Let X and Y be continuous random variables with joint probability density function fx,y(x,y) 0, otherwise (a) Compute the value of k that will make f(x, y) a legitimate joint probability density function. Use f(x.y) with that value of k as the joint probability density function of X, Y in parts (b),(c).(d),(e (b) Find the probability density functions of X and Y. (c) Find the expected values of X, Y and XY (d) Compute the covariance Cov(X,Y) of X...
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).
Let X and Y be continuous random variables with joint distribution function: f(x,y) = { ** 0 <y < x <1 otherwise What is the P(X+Y < 1)?
55. Let X and Y be jointly continuous random variables with joint density function fx.y(x,y) be-3y -a < x < 2a, 0) < y < 00, otherwise. Assume that E[XY] = 1/6. (a) Find a and b such that fx,y is a valid joint pdf. You may want to use the fact that du = 1. u 6. и е (b) Find the conditional pdf of X given Y = y where 0 <y < . (c) Find Cov(X,Y). (d)...
Question 4: Let X and Y be two discrete random variables with the following joint probability distribution (mass) function Pxy(x, y): a) Complete the following probability table: Y 2 f(x)=P(X=x) 1 3 4 0 0 0.08 0.06 0.05 0.02 0.07 0.08 0.06 0.12 0.05 0.03 0.06 0.05 0.04 0.03 0.01 0.02 0.03 0.04 2 3 foy)=P(Y=y) 0.03 b) What is P(X s 2 and YS 3)? c) Find the marginal probability distribution (mass) function of X; [f(x)]. d) Find the...
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]