Write out a sequence of random variables {Xn}, n=1,2,…such that Xn converges to 0 in probability but {E(Xn), n=1,2,…} does not converge to 0.
Prove it.
Write out a sequence of random variables {Xn}, n=1,2,…such that Xn converges to 0 in probability...
Prove that if Xn is a sequence of random variables that converge to a limit X almost surely, then Xn converges to X in probability. Give an example to show that convergence in probability does not imply almost sure convergence. Suppose X bar is the mean of random sample of size 100 from a large population with mean 70 and standard deviation 20. Without use CLT , give the estimated probability P(65<x<75).
(1) Consider the probability space 2 [0, 1. We define the probability of an event A Ω to be its length, we define a sequence random variables as follows: When n is odd Xn (u) 0 otherwise while, when n is even otherwise (a) Compute the PMF and CDF of each Xn (b) Deduce that X converge in distribution (c) Show that for any n and any random variable X : Ω R. (d) Deduce that Xn does not converge...
9. In many cases where a sequence of random variables converges in probability to some b, this b will be either the expected value or the limit of the expected values of the variables. However, this is not generally true. (a) Consider a sequence of random variables where for each n, xn comes from this distribution with P(Xn = n) = 1/n and P(Xn = 0) = 1 - 1/n. Find limn+ E(Xn). (b) Find the value b such that...
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.
1. A sequence of random variables Xn satisfy Xn _>X in probability and E(Xn) -> E(X) for some random variable X (a) Show that E([X, - X|) -> 0 if Xn >0 for all n (b) Find a counterexample satisfying E(X,n - X) A0 if X are not non-negative. 1. A sequence of random variables Xn satisfy Xn _>X in probability and E(Xn) -> E(X) for some random variable X (a) Show that E([X, - X|) -> 0 if Xn...
Let X1, X2,...be a sequence of random variables. Suppose that Xn?a in probability for some a ? R. Show that (Xn) is Cauchy convergent in probability, that is, show that for all > 0 we have P(|Xn?Xm|> )?0 as n,m??.Is the converse true? (Prove if “yes”, find a counterexample if “no”)
Let a sequence of random variables Un converge in probability to c and the sequence of random variables Vn converge in distribution to d. Show that Un+Vn converges in probability c + d.
4. Let Xi, X2,... be uncorrelated random variables, such that Xn has a uniform distribution over -1/n, 1/n]. Does the sequence converge in probability? 5. Let Xi,X2 be independent random variables, such that P(X) PX--) Does the sequence X1 +X2+...+X satisfy the WLLN? Converge in probability to 0?
Consider a sequence of random variables X1, . . . , Xn, . . .where for each n, Xn ∼ t distribution. Apply Slutsky’s Theorem to show that as the degrees of freedom go to infinity, the distribution converges to a standard normal. (a) Let V1, . . . , V_n, . . . be such that Vn ∼ Chi Sq, n df. Find the value b such that V/n in probability −→ b. (b) Letting U ∼ N(0, 1),...
(Stochastic process and probability theory) Let Xn, n > 1, denote a sequence of independent random variables with E(Xn) = p. Consider the sequence of random variables În = n(n-1) {x,x, which is an unbiased estimator of up. Does (a) in f H² ? (6) ûn 4* H?? (c) în + k in mean square? (d) Does the estimator în follow a normal distribution if n + ?