Suppose X1,…,Xn,…are identically distributed with mean E(X1)=μ<∞ and Var(X1)=σ2<∞. In addition, we assume that Cov(Xk,Xk+1)=0 for k=1,2,… but Cov(Xk,Xj)=0 whenever ∣k−j∣≥2. (a) Find the limiting distribution ofXˉn=n−1i=1∑nXiasn→∞. (b) Find the limiting distribution of Zn=∑nnXin+∑n=1nXiX,eias n→∞. (c) LetY1,Y2… be i.i.d random variables with mean 0 and variance 1 . Additionally, letXk=Yk+Yk+1 for ,k≥1. Find the limiting distribution of Xˉn⋅

Suppose X1,…,Xn,…are identically distributed with mean E(X1)=μ<∞ and Var(X1)=σ2<∞. In addition, we assume that Cov(Xk,Xk+1)=0 for k=1,2,… but Cov(Xk,Xj)=0 whenever ∣k−j∣≥2. (a) Find the limiting distribution ofXˉn=n−1i=1∑nXiasn→∞. (b) Find the limiting distribution of Zn=∑nnXin+∑n=1nXiX,eias n→∞. (c) LetY1,Y2… be i.i.d random variables with mean 0 and variance 1 . Additionally, letXk=Yk+Yk+1 for ,k≥1. Find the limiting distribution of Xˉn⋅

A First Course in Probability (10th Edition)

10th Edition

ISBN:9780134753119

Author:Sheldon Ross

Publisher:Sheldon Ross

Chapter1: Combinatorial Analysis

Section: Chapter Questions

Problem 1.1P: a. How many different 7-place license plates are possible if the first 2 places are for letters and...

See similar textbooks

Similar questions

Let 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
In bacterial counts with a haemacytometer, the number of bacteria per quadrat has a Poisson distribution with probability mass function f(x), where f(x) = θ x e −θ/x! and θ is to be estimated. If there are many bacteria in a quadrat, it is difficult to count them all, and so the only information recorded is that the number of bacteria exceeds a certain limit c, a large positive integer. In a random sample of n quadrats, it was.
Let X1, .... Xn be a random sample from a population with location pdf f(x-Q). Show that the order statistics, T(X1, ...., Xn) = (X(1), ... X(n)) are a sufficient statistics for Q and no further reduction is possible?
Suppose that X is a continuous unknown all of whose values are between -3 and 3 and whose PDF, denoted f , is given by f ( x ) = c ( 9 − x^2 ) , − 3 ≤ x ≤ 3 , and where c is a positive normalizing constant. What is the variance of X?
LetX1,X2,...,Xn be a sequence of independent and identically distributed random variables having the Exponential(λ) distribution,λ >0, fXi(x) ={λe−λx, x >0 0, otherwise Define the random variable Y=X1+X2+···+Xn. Find E(Y),Var(Y)and the moment generating function ofY.
If X1, X2, ... , Xk have the multinomial distribution of Definition 8, show that the mean of the marginal distribu-tion of Xi is nθi for i = 1, 2, ... , k.
2) The time between successive customers coming to the market is assumed to have Exponential distribution with parameter l. a) If X1, X2, . . . , Xn are the times, in minutes, between successive customers selected randomly, estimate the parameter of the distribution. b) b) The randomly selected 12 times between successive customers are found as 1.8, 1.2, 0.8, 1.4, 1.2, 0.9, 0.6, 1.2, 1.2, 0.8, 1.5, and 0.6 mins. Estimate the mean time between successive customers, and write down the distribution function. c) In order to estimate the distribution parameter with 0.3 error and 4% risk, find the minimum sample size.
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-Y2
Consider two securities, the first having μ1 = 1 and σ1 = 0.1, and the secondhaving μ2 = 0.8 and σ2 = 0.12. Suppose that they are negatively correlated,with ρ = −0.8. Denote the expected return and its standard deviation as functions of π byμ(π ) and σ (π ). The pair (μ(π ), σ (π )) trace out a curve in the plane as πvaries from 0 to 1. Plot this curve in R.
Suppose that n observations are chosen at random from a continuous pdf fY(y). What is the probability that the last observation recorded will be the smallest number in the sample? I asked this question earlier today, but didn't quite understand all of the response. P(y1<=yn)p(y2<=yn) and so on was used, but shouldn't the yn be listed first in the inequality since we want to know if yn is the smallest?
If X is a random variable that is uniformly distributed between -1 to 1, find the PDF √|X| of and pdf of -ln|X|
Suppose that a study of a certain computer system reveals that the response time, in seconds, has an exponential distribution with density curve f(x) = (1/3)e(-x/3) for x > 0 and f(x) = 0 otherwise. What is the probability that response time exceeds 5 seconds? What is the probability that response time exceeds 10 seconds?