Question

Problem 3 (40 pts) Two random variables X and Y are jointly distributed, based on the equation JXY (1, Y) |xy (,y) = } : (1,Y) E D 10 (, y) &D (1) where D is the region illustrated below. 1. Find k. Answer: k = } 2. Find expressions for, and sketch, the marginal distributions fx(x) and fy(y) Answer: plomba 0<<1 1<r < 2 2 < < Sy() – 24,2 y = [0, 1] 3. Find E[XY]. Answer: E[XY] = 4

please show steps to find the given answers.

Answer 1

SS f(x, y) = 1 ay the integral Here we split into 3 part. dx + k dy da=1 9-42=X-14 42-4, 2-t Ilk dy dx + J J K 0 0 22- 4-02-2 1 X=34+2 k's 91% dx kif [9]ldir ki cum der! fy-x-2 it's x dx + k lide + k/6 - x2) dx = 11 *[22].' + ge + [ - ] =/ k+k+ k 6-a) – 4) =1 3 + 3k - 3 - 1 3k=1 K21/

a) fx (x ) = J f(x,y) dy Here we have 3 region. osxsl a f(x) = s f(x,y) dy = 3 1/3 dy ty] = x/ losx 52 f(x) - I f(x,y) dy = yes [ 47.

2XLS f(x) = S f(x,y) dy 붕 대 - [1 ]

23 fa 0 1< 112 5 - 277 24 25 = (9) دو ( 23 ) 32 - علميا 1 - 21+2 ديرعلا = = و 2 و3 و وه = ن

51 2 x 2 CCXY ) = f f fw, y) dy da + 1 xq dy dx + } $ xy dy do + J & Jaydy he - 2 x [ 22 23 dx e f f (471 4 2 2a [ 4.7 x ' + Brela je to fact-030 *jdz - .-) ++ – 6221) Bazar by the nut ja en orodeto stytt tó (Z - reception - +++25-4 - w 37 = a + 4 tal-q-27