Question

# 6 If two random variables have the joint density f(x, y)=59 y?) for 0<x<1, 0<y<1 0 elsewhere a. Find the probability that 0.2 X<0.5 and 0.4<Y<0.6. b. Find the probability distribution function F(x, y). c. Are x and y independent?

Answer 1

Let two random variables have the joint density function as f(x) = (x+42) 104441, oayas ffrey) dy dx 0.2 0.4 0.5 0.6 O 62-492) diy olx 0.2 0.4 olh ceph 0.6 on

. alos 5 € +19) dbe alos uplo nlos $ 16+52- (tas + 49) ) - (94103] 1 % -16] bel. Wanlos z (6025 – 392) 25 100 633 25x50 633 = 1250 = 0.5064

Hence, p(0.2<x<0.5, 0.444406) : 0.5064 Peug) - S trongyal dhe ac E gloi ST (2442) dy dx ala ala S by 4 * 3) da alo Hencen Fory)= ay (314+242) where, osx<1,OLYLI

fen: S'fwy) dy = "Set 3)dly - f(x)= 3 (1+33) - 0<x</ EC) = 5* fc) dx = = Sx (192) obe - 3 S (+372) dx = 2 (34* *8]. fy) = f(xy) dx = $ ( 42) ox = 3 (1 +242) f(Y) = 31+242) : 0<44

E(M)= Sufaidy. && (+25) dy S6 1947) de EGY): S. 984 504) obedy E(XY) = Slayfory dedy Stky (xyzy olady = & Shiny + x43) dx dy

= $(244343) dy + ( 42424) Cor (x, y) = E(XY) – EX) E(Y) EZ - os - 35–36Too Cov(x,y) 70 Hencen x and y are not independent. 100