Question

1. Suppose the joint density of X and Y is given by f(x,y) = 6e^-3x-2y, if 0 < x < inf., 0 < y < inf, 0 elsewhere.

Part A,

Find P( X < 2Y)

Part B,

Find Cov(X,Y)

Part C,

Suppose X and Y have joint density given by

f(x,y) = 24xy, when 0<= x <=1, 0 <= y <=1, 0 <= x+y <=1, and 0 elsewhere.

Are X and Y independent or dependent random variables? why?

Answer 1

Ocycas. 1x,y) = 6e- 3x-24 OCXL0 ow. P(XL24) = 12 66-3x-24 ayda to bez (py dx) ay • To Ge-zy wolf 0-32 1 2 = -2 100 1-2y (e-3x2y e) 2 1*-_-2348-09-1) dy Jo JO = 2 hou (e-33 --e-89) dy 1-13 28 2 P( x2 2Y) = 3 | (8) . G x PDF f (x) = . e-3x-2y dy 6 - - 6e-30 e 24 day = e e 3t = (3 e 3) f(x) = f 3e-32 10 ol

PDF of his f(1) = 60 c-32-24 de = 6e-2 Toe de flu) = (2824) Since fony) = f(x). fl) - 6e-32-24 = So, X and Y are (20 32) (26-27) independent. So cov (X,Y)=0. fixy) = 24 ay OLRLI, ocys 2t9u. PDF of x from = 24 ay ay 3 24 2 (y2 90 122 (1-x2 ocxy PDF of 4 is ly fly) = 24 ay ay = 244c1-42 2 fly) 12 x (1 y ² f(x) = frafly) 024-21 Since. 24 xy & 12311-202 12 4 (1-4)2 So, X and Y are not independent. They are dependent