Let X and Y be independent random variables each having the uniform distribution on [0, 1]. Let U = min{X, Y) and V = max(X, Y). Find E(U), and hence calculate cov(U, V).
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- Let X and Y be two continuous random variables having joint pdffX,Y (x, y) = (1 + XY)/4, −1 ≤x ≤1, −1 ≤y ≤1.Show that X ^2 and Y ^2 are independent.Let random variables X and Y have the joint pdf fX,Y (x, y) = 4xy, 0 < x < 1, 0 < y < 1 0, otherwise Find the joint pdf of U = X^2 and V = XY.For any continuous random variables X, Y , Z and any constants a, b, show the following from the definition of the covariance:
- 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.Use the moment generating function technique to solve. Let X1, . . . , Xn be independent random variables, such that Xi ∼ Exponential(θ), for i =1, . . . , n. Find the distribution of Y = X1 + · · · + Xn.Let X and Y be continuous random variables with joint distribution function, F (x,y). Let g (X,Y) and h (X,Y) be functions of X and Y. PROVE Cov (X,Y) = E[XY] - E[X] E[Y]
- Let X1, . . . , Xn i.i.d. U([θ1, θ2]), i.e., X1, . . . , Xn are independent and follow a uniform distribution on the interval [θ1, θ2] for θ1, θ2 ∈ R and θ1 < θ2. Find an estimator for θ1 and θ2 using the method of moments.Let X and Y be independent random variables, and let α and β be scalars. Findan expression for the mgf of Z = αX + βY in terms of the mgf’s of X and YLet X1, X2, · · · , Xn be uniform i.i.d. random variables between [0, 1].Find the CDF and PDF of X = min(X1, X2, · · · , Xn)
- Let X1, . . . , Xn be independent random variables, such that Xi ∼ Exponential(θ), for i =1, . . . , n. Find the distribution of Y = X1 + · · · + Xn.X is an exponential random variable with λ =1 and Y is a uniform random variable defined on (0, 2). If X and Y are independent, find the PDF of Z = X-Y2Let X, Y, and Z be jointly distributed random variables. Prove that Cov(X + Y, Z) = Cov(X, Z) + Cov(Y, Z).