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Let (X, Y) denote a uniformly chosen random point inside the unit square
[0, 2]2 = [0, 2] × [0, 2] = {(x,y) : 0 ≤ x, y ≤ 2}.
What is the probability P(|X − Y| ≤ 1/2)?
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- Find the moment-generating function of the continuous random variable X whose probability density is given by f(x) = 1 for 0 < x < 1 0 elsewhere and use it to find μ’1,μ’2, and σ^2.Use the moment generating function to solve. Let X1, . . . , Xn be independent random variables, such that Xi ∼ Poiss(λi), for i = 1, . . . , n.Find the distribution of Y = X1 + · · · + Xn.For any continuous random variables X, Y , Z and any constants a, b, show the following from the definition of the covariance:
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- The density of a random variable X is f(x) = C/x^2 when x ≥ 10 and 0 otherwise. Find P(X > 20).Suppose that the random variables X, Y, Z have multivariate PDFfXYZ(x, y, z) = (x + y)e−z for 0 < x < 1, 0 < y < 1, and z > 0. Find (a) fXY(x, y), (b) fYZ(y, z), (c) fZ(z)Let Y be a discrete random variable. Let c be a constant. PROVE Var (Y) = E (Y2) - E (Y)2
