Let Ui,U2Ube independent Unif-2,0) random variables and X)U,U). Prove that X(u) converges in probability to -2.
6. Let Ui, U,Un be independent Unif-2,0) random variables and Xa)in(U, U2, .., Un). Prove that X(a) converges in probability to -2
Prove that a sequence of random variables X1, X2, ... converges in probability to a constant μ if and only if it also converges in distribution to μ. 5. Prove that a sequence of random variables X1, X2,... converges in probability to a constant p if and only if it also converges in distribution to u.
2. Let X and Y be two independent discrete random variables with the probability mass functions PX- = i) = (e-1)e-i and P(Y = j-11' for i,j = 1, 2, Let {Uni2 1} of i.i.d. uniform random variables on [0, 1]. Assume the sequence {U i independent of X and Y. Define M-max(UhUn Ud. Find the distribution
(10 marks) Let X1, X2,... be a sequence of independent and identically distributed random variables with mean EX1 = i and VarX1 = a2. Let Yı, Y2, ... be another sequence of independent and identically distributed random variables with mean EY = u and VarY1 a2 Define the random variable ( ΣxΣ) 1 Dn 2ng2 i= i=1 Prove that Dn converges in distribution to a standard normal distribution, i.e., prove that 1 P(Dn ) dt 2T as n >oo for...
D. Let Xi, X2,. be independent random variables from a uniform distribution over the interval [0, 1]. Prove that the sequence X+XX. converges in probability and find the limit
There are two independent Bernoulli random variables, U and V , both with probability of success 1/2. Let X=U+V and Y =|U−V|. 1) Calculate the covariance of X and Y 2) Explain whether X and Y are independent or not 3) Identify the random variable expressed as the conditional expectation of Y given X, i.e., E[Y |X].
15. Suppose Ui ~ iid Unif(0, 1) for n = 6. Let X = U(1), Y = U(6), and W = X/Y. Find: ~Ll b) Fw(w) c) E(W) d) Var(W)
Let X1, X2, · · · be independent random variables, Xn ∼ U(−1/n, 1/n). Let X be a random variable with P(X = 0) = 1. (a) what is the CDF of Xn? (b) Does Xn converge to X in distribution? in probability?
The random variables X and Y are independent with exponential densities fx (x) = e-"u(x) (a) Let Z = 2X + and w =-. Find the joint density of random variables Z and W (b) Find the density of random variable W (c) Find the density of random variable Z The random variables X and Y are independent with exponential densities fx (x) = e-"u(x) (a) Let Z = 2X + and w =-. Find the joint density of random...
Let x and x, be independent random variables with Mean u and variance o2. Suppose that we have two estimators Of u : A @= X1 + X2 2 and ©2 = X, +3X2 2 (a) Are both estimators unbiased estimators of u? (b) What is the variance of each estimator?