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
Let X and Y have join density 6 f(x, y) =-(x + y)2, 0 < x < 1, 0 < y < 1
Let X N(1,3) and Y~ N(2,4), where X and Y are independent 1. P(X <4)-? P(Y < 1) =? 4、 5, P(Y < 6) =? 7, P(X + Y < 4) =?
Let X, Y be random variables with f(x, y) = 1,-y < x < y, 0 < y < 1. Show that Cov(X,Y) = 0. Are X, Y independent?
4. (Sec. 5.2, 00) Let X and Y be continuous rvs with the joint f(x, y) = 2(x+y), for 0 <y <r <1 and 0 otherwise. (a) Find E(X+Y) and E[X - Y) (b) Find E[XY] (c) Find E[Y|X = x) and E[X Y = y). (d) Find Cov[X,Y]
8. Let f (x) e, 0 > 0; x> 0 (1 1 +e (a) Show that f(x) is a probability density function (b) Find P(X> x) (c) Find the failure rate function of X
8. Let X and Y be a random variable with joint continuous pdf: f(x,y)- 0< y <1 0, otherwise a. b. c. Find the marginal PDF of X and Y Find the E(X) and Var(X) Find the P(X> Y)
1. Verify that fxr (x,y) -2e-x-y 0 < x < 00, x < y < is joint probability density function 2. Compute the probability that X < 1.and Y < 2.
solve for c such that f(x,y) is a valid density function. Seiten f(x, y) = 1<x<y <3 otherwise 0,
(2.2) Let a be a real number with 1<a< 2. Put f(x) = Q +r 1+2 (a) Show that f maps (1, 0) into (1, 0). (b) Show that f is a contraction on [1, ) and find its fixed point.