Prove that a sequence of random variables X1, X2, ... converges in probability to a constant...
(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...
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 X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following convergence results independently (i.e, do not conclude the weaker convergence modes from the stronger ones). d Yn 0. a. P b.Y 0. L 0, for all r 1 Yn C. a.s d. Y 0. Let X1, X2, X3, . be a sequence of i.i.d. Uniform(0,1) random variables. Define the sequence Yn as Ymin(X1, X2,,Xn) Prove the following...
8. Let X1, X2,...,X, U(0,1) random variables and let M = max(X1, X2,...,xn). - Show that M. 1, that is, M, converges in probability to 1 as n o . - Show that n(1 - M.) Exp(1), that is, n(1 - M.) converges in distribution to an exponential r.v. with mean 1 as n .
Consider two sequences of random variables X1, X2,... and Y1, Y., .... Suppose that Xn converges to a and Y, converges to b with probability 1. Show that X, + Y, converges to a+b, with probability 1. Next, we assume that the random variables Y cannot be equal to zero, show that X/Y, converges to a/b with probability 1.
Central Limit Theorem: let x1,x2,...,xn be I.I.D. random variables with E(xi)= U Var(xi)= (sigma)^2 defind Z= x1+x2+...+xn the distribution of Z converges to a gaussian distribution P(Z<=z)=1-Q((z-Uz)/(sigma)^2) Use MATLAB to prove the central limit theorem. To achieve this, you will need to generate N random variables (I.I.D. with the distribution of your choice) and show that the distribution of the sum approaches a Guassian distribution. Plot the distribution and matlab code. Hint: you may find the hist() function helpful
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...
Let X1, X2,..., X, be n independent random variables sharing the same probability distribution with mean y and variance o? (> 1). Then, as n tends to infinity the distribution of the following random variable X1 + X2 + ... + x, nu vno converges to Select one: A. an exponential distribution B. a normal distribution with parameters hi and o? C a normal distribution with parameters 0 and 1 D. a Poisson distribution
two random variables x1 and x2 have a joint probability density function f(x1,x2)={x1+x2, 0<x1<1, 0<x2<1 0, otherwise what is the marginal distribution of x1 and x2
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.