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

Related questions

Q: 1. Let X1, Xn be i.i.d. random variables with a pdf .... f(x) = = e-(x-0), 10, x > 0, elsewhere for…

A: The objective of the question is to compute the cumulative distribution function (CDF) of Y and Zn,…

Q: Let X1, X2,...,X, be a set of independent random variables each following the distribution with pdf…

A: Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Let Y₁, Y2,..., Yn denote a random sample of size n from a population with a uniform distribution on…

A: To find the expected value of the smallest order statistic Y(1) from a uniform distribution on the…

Q: 8. Let {X, r≥ 1} be independent with the distribution of the preceding Un = n¹"=1 Xr, and Tn = n¹Un.…

Q: Let X and Y be statistically independent random variables with Var(X) = 4 and Var(Y) = 10. We do not…

A: The provided information is as follows:. and are independent random variables..

Q: Let X₁, Xn be a random sample of size n from the U(0, 0) distribution, where > 0 is an unknown…

A: Given that be a random sample of size from the distribution, where is an unknown parameter.

Q: Let X1,..., Xn be iid pdf (log(æ)-®1)² 262 1 f(x) -00 < x < o0, and unknown parameters 01 and 02.…

A: Let X1,X2, X3...be iid with pdf, f(x)=1x2πθ2e-(log(x)-θ1)22 θ2 -∞<x<∞ The…

Q: Suppose X has a Poisson distribution with parameter A and suppose the conditional distribution of Y…

Q: 27. If X is uniformly distributed over (0, 1) and Y is exponentially distributed 1, find the…

A: X is uniformly distributed over (0, 1) and Y is exponentially distributed with parameter .

Q: Set n > 2. Let X1, X2, ..., Xn denote a random sample from a uniform distribution on (0, 1) i.e.…

A: Given that, Xi ~U(0, 1) i= 1,2,3......n We have to find E(R) where R=Y(n) - Y(1)

Q: 5. Let X₁, X2,..., Xn are random samples from the pdf f(x; 0) = 0/x², 0 < 0≤x. a) Find the mle of 0.…

A: f(x;θ)=θx2;0<θ<x

Q: Suppose X1, . . . , Xn are i.i.d. from a continuous distribution with p.d.f. fθ(x)=1/2(1+θx)if…

A: X1, . . . , Xn are independent and identically distributed (i.i.d) random variables with pdf…

Q: 4. (20%) Customers arrive at a service facility according to a Poisson process of rate A=2…

A: a) E(X(4)X(5)|X(1) = 3) = E(X(4)|X(1) = 3) * E(X(5)|X(1) = 3) = 9 * 11 = 99b) E(X(1) + X(3) | X(2) =…

Q: Let xy X2 Xn be a random sample from a distribution ...... with p.d.f. f (x, 0)= e- (x – 0); 0< x< ∞…

A: ANSWER: Form the given data,

Q: 20. A continuous random variable X has CDF given by if x <1 F(x) ={2(x – 2 + 1/x) if 1<x<2 if 2 < x…

A: As Prof. A. K. Giri defined, "Probability Density Function: A continuous function f(x) is said to be…

Q: Find the mean and standard deviation for each uniform continuous model U(1, 13) U(70, 200) U(3, 92)

A: Obtain the mean and standard deviation for the uniform continuous model U(1,13). Obtain the mean of…

Q: Consider the continuous random variable X whose pdf is given by f(2)=√15.50(2). (a) Find the value…

A: Note: Hi there! Thank you for posting the questions. As there are multiple sub parts, according to…

Q: 1. Let the random variable Y have pdf as f(y\A) = e®=e=t(e-1), y>0,1>0. Show that W = e – 1] ~ x²…

Q: Marginal Distributions For the discrete case: so that for all x₁ E Dx₁. F=Σ Fx (x1) = Σ PX1X2(W1W2)…

A: Here pX1,X2x1,x2 denotes the joint probability mass function (PMF) at X1=x1 and X2=x2 and pX1x1…

Q: nstants аnd 9. Fx (x) = [1 – a. e-x/b]u (x) is a valid distribution funetion. u (x) is an unit step…

A: Answer: For the given data

Q: 5. Suppose that X~U(a, b), where U represents a continuous uniform distribution; that is, the pdf of…

A: X is continuous uniform distribution in the interval (a,b). X ~ U(a,b) So, Variance of X, Var(X) =…

Q: A statistic is denoted by T(X1,.., X„) or simply T(X) 14.- Let X1, X2 be a random sample from the…

A: “Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: Let X~ Unif(0, 2) and Y~ Bi(2, 0.5). If Z is a mixture of X and Y with equal weights, graph the…

A: Given that X~Unif0,2Y~Bi2,0.5 fX=12-0, 0≤X≤2FX=X-02-0, 0≤X≤2 =X2,…

Q: Suppose that the elements of X = [X₁, X2, ..., Xn]' form a random sample from a Poisson (A)…

A: From cramer- rao lower bound we know that - Var(λ⏞(x)) ≥ {ψ'(θ)}2I(θ) ......(1) Where symbols have…

Q: fp(p) = 12 p²(1-p), 0<p<1. (a) Calculate E(X). (b) Obtain Var (X). (c) Find the marginal pmf of X.

Q: Let X₁,..., Xn be a random sample of size n from the U(0, 0) distribution, where > 0 is an unknown…

A: Given that be a random sample of size from the distribution, where is an unknown parameter.

Q: Let X and Y be two continuous random variables with the joint probability distribution function:…

Q: Suppose Y₁, Y₂,..., Yn from Poisson distribution with Y;~ Poisson(₁) ‚μ; depends on x;. Give the…

A: Let Y1,Y2,....,Yn follows Poisson distribution with Yi~Poisson(μi). The probability density function…

Q: Let X be a random variable with pdf given by 1 f(x)= π(1+x²)' Find Mx (t). -∞ < x <∞.

Q: Suppose that that the RV X has the Gamma(2,β ) distribution for some β > = 0, E(X) = 2β and V (X) =…

A: Given that the RV X has the distribution for some , and .Given , the RV Y has the distribution.

Q: Let X1 be the sample mean of a random sample of size n from a Normal(µ, o7) population and X2 be the…

A: Let X1,X2,......,Xn be a random sample from a normal population N(μ,σ2). The distribution of sample…

Q: Consider the continuous random variable X whose pdf is given by f(z)=(a-³) LOSISI(I). (a) Find the…

A: Note: Hi there! Thank you for posting the questions. As there are multiple sub parts, according to…

Q: distributed random variables with Var(X1) < . Show that 1 n(n +1) jX; →, EX1. j=1 Note. A simple way…

Q: Let X₁,, Xn be an iid random sample from a Uniform(0, 0), where > 0. Let X(n) = max{X₁,, Xn} (a)…

A: Let X1, X2........,Xn be random sample from a UniformIts PDF is,Let Xn=max {X1, X2........,Xn}.

Question

100%

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→∞.

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⋅

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1: part 1

Step 2: part 2

Step 3: part 3

Solution

Trending now

This is a popular solution!

Step by step

Solved in 4 steps

SEE SOLUTION Check out a sample Q&A here

Similar questions

1. Suppose we have a random sample X₁, X2, ..., Xn from an Exponential distribution with mean 1/A. Suppose we want to estimate the mean 1/A. One estimator for 1/X is T₁ = X. Of interest is to note that the minimum of X₁, X2,..., Xn, say X(1), has an Exponential distribution with mean (n)-¹. (a) Show that T₁ is an unbiased estimator of 1/A. (b) Find the constant a such that T₂ = aX(1) is an unbiased estimator of 1/X. (c) Since T₁ and T₂ are both unbiased we prefer the estimator with smaller variance. Which of the estimators T₁ and T₂ would you choose for estimating the mean 1/X?
Suppose Xn is the sample mean of a random sample of size n from a distribution that has a Gamma(2,1/2) (where a = 2, B = }) pdf f(x) = 4xe-2a, 0 < x < ∞, zero elsewhere. Use the Central Limit Theorem to deduce that the random variables Vn(Xn – 1) converges in distribution to N(0, 1/2).
7. a) Suppose that X is a uniform continuous random variable where 0
3. Let X1, and E- X; are independent. , Xn be iid with exponential distribution with mean 0. Show that (X1+X2)/ E=1 X;
Let X1,..., Xn be a random sample from a uniform distribution on the interval [20, 0], where 0 0. Let X(1) < X(2) <...< X(n) be the order statistics of X1, ..., Xn.
1. If X,X2, .,X, forms a random sample of size n from a population with pdf f(x; 0,8) = for x>0 elsewhere Find estimators for 8 and 0 by the method of moments.
Which of the following is the correct sum of squared residuals to be minimized to find tne fit f(x) = £ (2r-1)"* - using a data set ((xo, Yo), (x1,y1), --, CKn. Yn)} ? A.) Eo(In(y;) + a – b2x; + 1)² B.) E-(In(y;) + In a + b In(2x; – 1))² D.) E7-o(In(y;) – In a – b In(2x; – 1))² E.) E-o(In(y;) + In a – b2x; + 1)²
The continuous random variable X on [-1,1] has pdf given by fx(x) = k(3x + 5) for x = [1,1] where k is a constant that you are going to determine. fx(x) = 0 outside the interval [-1,1] Hint: consider Calculate the following: i. k ii. E(X) iii. Var(X)
page 328 5.2.2
3) Assume that a random variable X has moment generating function as follows: e' + e²¹ M(t)=4-2e¹ ∞ Calculate the mean and variance of X. for t
If X1, X2,...,Xn is a random sample from a distribution having pdf of the form f(x; 0) = 0x°-1, 0 < x < 1, zero elsewhere, show that a best critical region for testing Ho : 0 = 1 against H1 : 0 = 2 is C = {(x1, 12, ..., Tn) : c< II-1 *i}. ----- DA --