5. (15 points) Let X and Y be independent exponential random variables with common mean 1/X, X,Y IN Exp(A) and consider

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5 15 Points Let X And Y Be Independent Exponential Random Variables With Common Mean 1 X X Y In Exp A And Consider 1 (57.9 KiB) Viewed 35 times

5. (15 points) Let X and Y be independent exponential random variables with common mean 1/X, X,Y IN Exp(A) and consider the (random) quadratic function f(t): { _ 2Xt+Y. 1 (5 points) Show that the probability that both roots of f(t) are real is given by P(X) = 1 - (TX)¹/² ¹/4 {1 – Þ ((A/2)¹/²)}, where is the standard normal c.d.f. 2 (5 points) Plot p(A) as a function of A. 3 (5 points) Suggest a Monte Carlo algorithm to approximate (1) by generating two independent uniform random variables U₁ and U2, U1₁, U₂ ~ Uniform(0, 1) and provide your estimate of (1) based on your algorithm.