Solutions and Remarks For Homework For Section 5.2, Page 354
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3. a) int_{0}^{1} int_{0}^{1} c(x^2 + 4xy) dx dy = 1  =>  c = 3/4

   b) P(X leq a) = int_{0}^{a} int_{0}^{1} f(x,y) dy dx = 3a^2(1+a/3)/4

   c) P(Y leq b) = int_{0}^{b} int_{0}^{1} f(x,y) dx dy

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5. f(x,y) = (lambda)(mu) e^(-lambda x - mu y)

   P(X geq 3Y) = int_{0}^{infty} int_{0}^{3x} f(x,y) dy dx

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9. f(s,t) = (lambda)^2 e^[-lambda(s+t)]

   f(x,y) dx dy = P(X \in dx, Y \in dy)
                = P(S \in dx, T \in dy) + P(S \in dy, T \in dx)
                =([lambda^2 e^(-lambda(x+y))]+[lambda^2 e^(-lambda(x+y))])dxdy



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11. f(y) = e^(-y); f(x) = 1, 0<x<1
                        = 0, ow

    a) E(X+Y) = E(X) + E(Y) = 1/2 + 1

    b) E(XY) = E(X) E(Y) = (1/2)(1)

    c) E[(X-Y)^2] = E(X^2 - 2XY + Y^2) = (1/3) - 1 + 2

    d) E(X^2 e^(2Y)) = E(X^2) E(e^(2Y))
                     = (int_{0}^{1} x^2 dx) (int_{0}^{infty}e^(2y) e^(y) dy

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