Let f(x, y) 2e-(x+y), x > 0, y > 0. Show that X, Y are independent....
Let f(x,y) = 12e-2(x+y), x > 0, y > 0. Show that X, Y are independent. What are the marginal PDFs of each?
Let f(x, y) = kxy, for 0 <x< 1 and 0 <y<1 and 0 elsewhere, a) Find k b) Find marginal pdfs. c) Are X and Y independent? d) Find P(X<0.5, Y>0.5).
Let X, Y be two independent exponential random variables with means 1 and 3, respectively. Find P(X> Y)
4. Let X, Y, and Z be independent random variables, each with the standard normal distribution. Compute the following: (a) P[X + Y> Z +2 (b) Var3x 4Y;
PROB5 Let U and V be independent r.v's such that the p.d.f of U is fu(u) = { 2 OSU< 27, otherwise. and the p.d.f'of2 is Seu, v>0, fv (v otherwise. Let X = V2V cos U and Y = 2V sin U. Show that X and Y are independent standard normal variables N(0,1).
2. Let the random variables X and Y have the joint PDF given below: S 2e-2-Y 0 < x < y < fxy(x,y) = { 0 otherwise (a) Find P(X+Y < 2). (b) Find the marginal PDFs of X and Y. (c) Find the conditional PDF of Y|X = r. (d) Find P(Y <3|X = 1).
4.3. Let X and Y be independent random variables uniformly distributed over the interval [θ-, θ + ] for some fixed θ. Show that W X-Y has a distribution that is independent of θ with density function for lwl > 1.
7. Let X1,... , Xn be iid based on f(x; 6) -22e-z?/e where x > 0. Show that θ=-yx? is efficient
2. Let the random variables X and Y have the joint PDF given below: 2e -y 0 xyo0 fxy (x, y) otherwise 0 (a) Find P(X Y < 2) (b) Find the marginal PDFs of X and Y (c) Find the conditional PDF of Y X x (d) Find P(Y< 3|X = 1)
PROBLEM 1 Let the joint pdf of (X,Y) be f(x, y)= xe", 0<y<<< a. Compute P(X>Y). b. What is the conditional distribution of X given Y=y? Are X and Y independent? c. Find E(X|Y = y). d. Calculate cov(X,Y).