Question

3. Chapter 6, Problem 9 The joint probability function of X and Y is given by f(x,y) =ý (x2 + **) osxs1 and 0 sys2 a. Verify that this is indeed a joint density function b. Compute the density function of X c. Find P(x >Y) d. Find P (Y > 1x <) e. Find E[X] f. Find E[Y]

Answer 1

3.

e a 6 A joint density will have an alaways positive measure over the joint support, and the integral over the point will equal to 1. the integral fasy) = f (rah + salt), olas, osyc2 Je l' (2+24) dx dy & Janda dy && is a dedy 461) 4+ $* y[Idy

14]6+ $+(91 - $ 1582-0)+ extes +6+ = 1 Cams b) the density function of is given by, Se copt met) dysosasi 4 | 3- + - 2X+X. , 0xxx! c) P(x> ") = 1 5 2 2 + xy) dydoc. و م رام

* * * 2 (1+ ) г = 0. 2012 олиб ) 4) PC У> 5) х» 2) - PC <3<2 о< x=) P(OLX2%) P(4942, 64x24)

26 3 269 = 0.1540 ans PCocx<») = %C4 x +x - 448 = 2 (2x²+a) da 2 11/56=0.1964 PCYS 41*<%) = 49/4148 = 0.78463 V/50

c) E(X) = f'x narejenal dewwilyafle de 1x de tant de - " + x = 5/7 = 0.7143(any) f) EY) = (y marginal density of y dy marginal density of y 163) = Life+ 2) da, 02452 N Un o N + 2 2 2 + 1 2 Os 462 EC) = 17 (3 + *) dy N - 2 x 2 + 1 4 3 = 1.2381 (ans)

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