Question

(3x, The joint density function of X and Y is given by 0 Sy sxs1 f(x, y) = 0, otherwise. a) Use the distribution function technique to find the distribution function of W = X-Y. For 50% of the points, you may use the transformation technique, which is longer. >) Find the probability density function of W. Find the expected value E(W). )

Answer 1

аш, 7 Q <1 о (0 0 1) Чо tmiys 1. ; o <<<< find dieloribution function of w fw (w) P[ We wl - PDX- Чs w РГуг иони =P <x< инч] — ЧЯ? Хя в к Chith chems x shj hixho = ф„Соня » 1) - ф (ч. 1) to find probability density con а) - Х- Ч X - 4 у X — — 0 + 0 let o=4 »» ЧЕ И Ч - 0 first andere derivative matrix ЭХ до эх ou 1 he δων he ine О J { tw,0 ) { (xy) - 15) Зу - 3 Cu+ а) , You ОҶИК) - < О 1 0 0° < | 2 0 и 3 | 'u 40 + ИЗ ОКю да О 7 И г wzi-u

Coll) qulos)- S+(,0) du UT Reu) T-W = J 3Twta] du 40 utw=1 2 (16) = 3 [wu +423 00 3 [ WC-6) +(1-633 -0-0) = 3 TW-W2 + 1 +w2-200 3 500-262 +1 +69-20] = 3 Ľ 1 - W2] 0 <wll 2 2 j ® (o) 3) E (W) s كيا 'w) do Rw s'w. gow) 3T160?] dw S'w-603 Jowo [w? ] حلم = w" ] w 4 = 318

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