Question

Need help with question 2 (not question 1)

1. Suppose that (X,Y) is uniformly distributed over the region {(x, y): 0 < \y< x < 1}. Find: a) the joint density of (X, Y); b) the marginal densities fx(x) and fy(y). c) Are X and Y independent? d) Find E(X) and E(Y). 2. Repeat Exercise 1 for (X,Y) with uniform distribution over {(x, y): 0 < \x]+\y< 1}.

Answer 1

1 2.) Lee S ={(44): 0 210)+14,41 :: Arca os S=4*3=2 (a) the joint density sum of (x,y) is $(14) = n-you Acom le 71745-1 71 D Олегоѕ Ѕ U SOABX1 (nUES 2 = 2 by <sco 1+1 16n20 (6) & Now marginal densitya fun of ox ir giver fxcm) = of (nu) dy when analo theme fx (n) = S t 2 dy = { [x] (*+1+ion) when OLULI then fx (2) = ^ [1-1-1+) = 1-7, obn2) .: fx (n) = x+1 El-n, o LALI Now é marginal densily oosy is given by fony) du 1/2 dy N-1 -157120 1 n itu an St -144 LO 2 -1-4 = HY ,-1+YLO I-Y when otył then fy (4) - Shan = 1-4, 0242 -12 YLO .fy (8)? 8+1 OLYLI CS Scanned with CamScanner

(c) clearly an annons fx (n) fy (0) + Fr, 121,8) = x,y are not independent. (&) Now B (²) Sn fx (m) dn (kan) du Se cun) de + fe con 0 f 2[W+231 [} - ofo-63 •4)3 +{4-3-0} - +*=0 Since xxx hartie same fun shen Ey) i also o E(Ⓡ) 2 E(Y)=0 IcScanned with CamScanner