Choose the all correct things Sn=X1+X2+...+Xn, Mn=Sn/n, Xi: iid RV 1. E[Mn]=nE[X], when Xi is iid...

Choose the all correct things

Sn=X1+X2+...+Xn, Mn=Sn/n, Xi: iid RV

1. E[Mn]=nE[X], when Xi is iid

2.VAR[Mn]=VAR[X]/root(n), when Xi is iid

3.As n goes to infinity, Mn's distribution approaches Gaussian. when Xi is iid

4.To change Sn to a distribution of an average of zero and a standard deviation of 1, it must be converted to Zn=(Sn-nE[X])/(root(n)*root(VAR[X]), when Xi is iid

Expert Solution

Therefore, option (3.) And (4.) are correct

orchestra answered 2 years ago

Let X = ( X1, X2, X3, ,,,, Xn ) is iid, f(x, a, b) =...

Let X = ( X1, X2, X3, ,,,, Xn ) is iid, f(x, a, b) = 1/ab * (x/a)^{(1-b)/b} 0 <= x <= a ,,,,, b < 1 then, find a two dimensional sufficient statistic for (a, b)

Let X1, X2, . . . , Xn iid∼ N (µ, σ2 ). Consider the hypotheses...

Let X1, X2, . . . , Xn iid∼ N (µ, σ2 ). Consider the hypotheses H0 : µ = µ0 and H1 : µ (not equal)= µ0 and the test statistic (X bar − µ0)/ (S/√ n). Note that S has been used as σ is unknown. a. What is the distribution of the test statistic when H0 is true? b. What is the type I error of an α−level test of this type? Prove it. c. What is...

Let X1, X2, · · · , Xn be iid samples from density: f(x) = {θx^(θ−1),...

Let X1, X2, · · · , Xn be iid samples from density: f(x) = {θx^(θ−1), if 0 ≤ x ≤ 1} 0 otherwise Find the maximum likelihood estimate for θ. Explain, using Strong Law of Large Numbers, that this maximum likelihood estimate is consistent.

Let estimator π(hat) = X(bar) for X1, X2, . . . , Xn, Xi ∼ Bernoulli(π)...

Let estimator π(hat) = X(bar) for X1, X2, . . . , Xn, Xi ∼ Bernoulli(π) Recall: P(X = x) = πx (1 − π)1−x , x ∈ {0, 1} E(X) = π V(X) = π(1 − π) a. Show that π(hat) is a Consistent estimator of π b. Find the Maximum Likelihood Estimator of π c. Show that π(hat) is a Minimum Variance Unbiased Estimator of π Please explain the answer in detail

Let X1, ..., Xn be iid with pdf f(x; θ) = (1/ x√ 2π) e(-(logx- theta)^2)...

Let X1, ..., Xn be iid with pdf f(x; θ) = (1/ x√ 2π) e(-(logx- theta)^2) /2 Ix>0 for θ ∈ R. (a) (15 points) Find the MLE of θ. (b) (10 points) If we are testing H0 : θ = 0 vs Ha : θ != 0. Provide a formula for the likelihood ratio test statistic λ(X). (c) (5 points) Denote the MLE as ˆθ. Show that λ(X) is can be written as a decreasing function of | ˆθ|...

Let X1,...,Xn be independent random variables,and let X=X1+...+Xn be their sum. 1. Suppose that each Xi...

Let X1,...,Xn be independent random variables,and let X=X1+...+Xn be their sum. 1. Suppose that each Xi is geometric with respective parameter pi. It is known that the mean of X is equal to μ, where μ > 0. Show that the variance of X is minimized if the pi's are all equal to n/μ. 2. Suppose that each Xi is Bernoulli with respective parameter pi. It is known that the mean of X is equal to μ, where μ >...

Consider the random sample X1, X2, ..., Xn coming from f(x) = e-(x-a) , x >...

Consider the random sample X1, X2, ..., Xn coming from f(x) = e-(x-a) , x > a > 0. Compare the MSEs of the estimators X(1) (minimum order statistic) and (Xbar - 1) Will rate positively

Let X1,…, Xn be a sample of iid random variables with pdf f (x; ?1, ?2)...

Let X1,…, Xn be a sample of iid random variables with pdf f (x; ?1, ?2) = ?1 e^(−?1(x−?2)) with S = [?2, ∞) and Θ = ℝ+ × ℝ. Determine a) L(?1, ?2). b) the MLE of ?⃗ = (?1, ?2). c) E(? ̂ 2).

1) Suppose X1,X2,...,Xn are iid Gamma(α,λ) random variables. a) Express the ﬁrst and third moments (µ1...

1) Suppose X1,X2,...,Xn are iid Gamma(α,λ) random variables. a) Express the ﬁrst and third moments (µ1 = E(X) and µ3 = E(X3)) as functions of λ and α. b) Solve this system of equations to ﬁnd estimators of ˆ λ and ˆ α as functions of the empirical moments. Note that the estimates must be positive. 2) Suppose that X1,X2,...,Xn are a iid Beta(a, a) random variables. That is, a beta distribution with the restriction that b = a. Using...

T/F 1) The function f(x) = x1 − x2 + ... + (−1)n+1xn is a linear...

T/F 1) The function f(x) = x1 − x2 + ... + (−1)n+1xn is a linear function, where x = (x1,...,xn). 2) The function f(x1,x2,x3,x4) = (x2,x1,x4,x3) is linear. 3) For a given matrix A and vector b, equation Ax = b always has a solution if A is wide

Question