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Exercise 6.34. Let (X,Y) be a uniformly distributed random point on the quadri- lateral D with vertices (0,0), (2,0), (1,1) a
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Sol: (a) Leo R(1,1) (ovla S P1010) (10)T (2,0) To find the point density of (x, y), compute the area of region D. Area of D =Marginal density function of x, I f(x,y) dy fx (0) If Oくんく 1 - for (x) = S z dy 3 Ž[4]. then If ocy < 2-x 1<x ca x fx (n) = 2Т? then о< 8 < 1 О< x < 2 - 2 - п, оо - - 1. 2 (2-8) 3 2 (2-1) o 4 441 Thus +, (3) else о SOL (6) ELх) — a f(x) dx 2 x da хy tyly) dy 3/3 (2-4) y dy 0 11 لا [-_] E(49) 4 9 sol (c) X and Y are are independent or not, To check whether compute Elx Y)Elxy) N f(x,y) xy dedy 2-4 2 ay dady - y ].9 y dy (2-ye dy u WIN 2 (4y - 4y² + y²) ay 11 ☆ [yyyyy $(?- 4 + 10 36 ECXY) 11 36

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