1. For pdf f (r, y) = 1.22, 0 < x < 1,0 < y < 2, z +y > 1, calculate: EY) and () E (X2)
NIS 4) The joint pdf of X and Y is 1, 0<x<1, 0<y< 2x, fx,8(8,y) = { 0, otherwise. otherwise. or 1 (Note: This pdf is positive (having the value 1) on a triangular region in the first quadrant having area 1.) Give the cdf of V = min{X, Y}. x
3. Let X has the following pdf: {. -1 <1 fx(a) otherwise 1. Find the pdf of U X2. 2. Find the pdf of W X
24. The joint cdf of (X,Y) is Find a) Joint pdf of (X, Y) b) Marginal pdf of X and Y c) PI(X s 1) n (Y s 1) d) PI(1 < X <3) n (1 <Y <2)] Page 4 of5
2. Let f(x,y) = e-r-u, 0 < x < oo, 0 < y < oo, zero elsewhere, be the pdf of X and Y. Then if Z = X + Y, compute (a) P(Z 0). (b) P(Z 6) (c) P(Z 2) (d) What is the pdf of Z?
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?
2. Suppose X and Y have the joint pdf fxy(x, y) = e-(x+y), 0 < x < 00, 0 < y < 0o, zero elsewhere. (a) Find the pdf of Z = X+Y. (b) Find the moment generating function of Z.
18. LetX and Y have joint pdf/(x, y) = eッ. 0 < x < y < oo and zero otherwise. (a) Find the joint pdf of S = X + Y and T-X. (b) Find the marginal pdf of T (c) Find the marginal pdf of S.
3 Let (X,Y) be a random vector with the pdf Se-(x+y), f(x,y) = e-(x+y) 122 (x, y) = 1 0, (x,y) E R otherwise. Find P{} <t}. In other words, find the PDF of the r.v. . Done in the class.
pectively, 3. Ajoint pdf is defined by (C(x + 2y), for 0 <x< 2, and 0 < y< 1, fx.r(x,y) = -{-4** 0. otherwise. a. Find the value of C. b. Find the marginal pdf of X alone. 9 c. Find the pdf of U = U = x+132 4. Consider n independent rvs XX2, ..., X, having the same distribution with a common variance a?. For any i = 1,2, ..., n, find Cov(x,- 8, 8), where 8 =...