Question

2. Let U and V be independent random variables, with P(U 1) 1/4 and P(U = -1) = P(V -1) 1) = P(V 3/4. Define X = U/V and Y = U V (a) Give the joint pmf of X and Y [4] (b) Calculate Cov(X,Y) [4]

Answer 1

2019-6-17 00:50 U and V ane indapendent Tamdo m variabies r(u1)PC) = 4 PC1FCVE) 4 amd r(Ve-) 3/4 Define valwes amd - ta Ke N (w 17 P(U) PCV)t P(u=-).P(v- - 1) independent - P(Uz) ' PCv) andom vaoi ao s P(U-I,V4) 3 . 10 1.1 4 . 4 P ( u- 1, ve-1) + P(02-t, ve 1) r(u-12. P(v-) P(-) P(vei) r( .5. (6 r(x-)6 16 Autine See that Y takes vawes Ne now 2,0-2 PCY=2)F(04va2) = PCU1, Ve) =u).r(ve) =1 r(y-o) r(uiveo) P(ue,v-)P(U1, ve) = P(UD f(ve- ) (U-) PCVE) 3 +3 16 -1 16

P (-2)P(v+v2-2) =r(u-1, Ve-1) P(uz-D PCV -) 3.2-9 4 16 16 PCY=2) 6 PlY:-2) 16 We tind out PCx=1, 2) rCu, VEi) P(x t, y-2)- r(u-lsva-) = this by taking iha ts com men 16 P(x=1) 1 n and f(y2) ia Ploct,vei) PLO7Y (x-1, Y-2)- e(*ニ-1) Y=ー)-。 f(x=-1Y o) 6 16 E (x) P(x) + (-1).P -1) - 6- 16 16 E (1) ナ(-2) P(y=-2) 2.rCy=2) +o.P(Y=0) 2 +o+ (-2) 9 16 I6 - 76 2 - N1 16 P(x- x, y9) (6) ELXY)= 17 1-2 P(x=1,Y2) . -) P(x-1,ye -a) + hoo-f(xz1,y-o) + C).2 9(-1,yo 2) +9-2) P-1, ye -2) 2019-6-17 00:51

2 1 1 6 +0-2.0 +2 0+ 0 76 =-1. l 6 16 E(x Y)- Elx)E() ', Cov(x, y) (09( = -1 + 4 2019-6-17 00:52 ilIli