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- Let {Xn} and {Yn} be sequences of random variables such that Xn diverges to ∞ in probability and Yn is bounded in probability. Show that Xn +Yn diverges to ∞ in probability.Prove that the rv Y=(tan^-1(X1/X2)) and Z=(X1^2+X2^2)^0.5 are independent. No distribution is given.Let X1 and X2 be two continuous random variableshaving the joint probability density f(x1, x2) = 4x1x2 for 0 < x1 < 1, 0 < x2 < 10 elsewhereFind the joint probability density of Y1 = X21 and Y2 = X1X2.
- Prove that cov(X, Y) = cov(Y, X) for both discreteand continuous random variables X and Y.LetXbea randomvariable havingthe uniformdistribution ontheinterval(θ,θ+1),θ∈R.Showthat Xis not complete.Suppose that we observe that X1, X2, . . . , Xn are iid∼ U(0, 1). Show that X(1)converges in probability to zero.
- Let XXX and YYY be uniformly distributed on [0,1][0,1][0,1] and independent. What is the expectation of the product XYXYXY, i.e., E[XY]E[XY]E[XY]? Provide at least two decimal places.4. Prove fx=x2+4 is not uniformly continuous on RLet X1,X2,... be a sequence of identically distributed random variables with E|X1|<∞ and let Yn = n−1max1≤i≤n|Xi|. Show that limnE(Yn) = 0
- 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.Let X be a random variable with pdf given by fX(x) = 1/[π(1 + x2)] for all real number x. Prove that X and 1/X are identically distributed by showing that they have the same probability distribution.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.