3. Given the following: N, the number of observations in a random sample. . S, the...
Please give detailed steps. Thank you.
5. Let {X, : i-1..n^ denote a random sample of size n from a population described by a random varaible X following a Poisson(θ) distribution with PDF given by θ and var(X) θ (i.e. you do not You may take it as given that E(X) need to show these) a. Recall that an estimator is efficient, if it satisfies 2 conditions: 2) it achieves the Cramer-Rao Lower Bound (CLRB) for unbiased estimators: Show that...
Please give detailed steps. Thank you.
5. Let {X1, X2,..., Xn) denote a random sample of size N from a population d escribed by a random variable X. Let's denote the population mean of X by E(X) - u and its variance by Consider the following four estimators of the population mean μ : 3 (this is an example of an average using only part of the sample the last 3 observations) (this is an example of a weighted average)...
Suppose that X1, X2,., Xn is an iid sample from the probability mass function (pmf) given by (1 - 0)0r, 0,1,2, 0, otherwise, where 001 (a) Find the maximum likelihood estimator of θ. (b) Find the Cramer-Rao Lower Bound (CRLB) on the variance of unbiased estimators of Eo(X). Can this lower bound be attained? (c) Find the method of moments estimator of θ. (d) Put a beta(2,3) prior distribution on θ. Find the posterior mean. Treating this as a fre-...
Let X1, X2, ..., Xn be a random sample with probability density
function
a) Is ˜θ unbiased for θ? Explain.
b) Is ˜θ consistent for θ? Explain.
c) Find the limiting distribution of √ n( ˜θ − θ).
need only C,D, and E
Let X1, X2, Xn be random sample with probability density function 4. a f(x:0) 0 for 0 〈 x a) Find the expected value of X b) Find the method of moments estimator θ e) Is θ...
Let X be a random variable with probability density function (pdf) given by fx(r0)o elsewhere where θ 0 is an unknown parameter. (a) Find the cumulative distribution function (cdf) for the random variable Y = θ and identify the distribution. Let X1,X2, . . . , Xn be a random sample of size n 〉 2 from fx (x10). (b) Find the maximum likelihood estimator, Ỗmle, for θ (c.) Find the Uniform Minimum Variance Unbiased Estimator (UMVUE), Bumvue, for 0...
- Suppose a random sample of size n is taken from the following distribution with a known positive parameter a. f(x;0,-) = a20 V 27797z exp 0; ; 0<x<00,0< < 0,0 < 8 < 00 elsewhere For this distruttore, the formats for mye or and x-a are respectively, Myo (1) = exp v{(1 - V1 –24*70)} for 1 < 2112 and exp{}(-VT - 2/0)} My-- (1) for 1 < ✓1 - 2t/0 2 Find the maximum likelihood estimators, 0 and...
QUESTION 2 Let Xi.. Xn be a random sample from a N (μ, σ 2) distribution, and let S2 and Š-n--S2 be two estimators of σ2. Given: E (S2) σ 2 and V (S2) - ya-X)2 n-l -σ (a) Determine: E S2): (l) V (S2); and (il) MSE (S) (b) Which of s2 and S2 has a larger mean square error? (c) Suppose thatnis an estimator of e based on a random sample of size n. Another equivalent definition of...
Show that the mean X bar of a random sample of size n from a distribution having probability density function f(x;θ)=(1/θ)e-(x/θ) , ,0 < x < ∞ , 0 < θ < ∞ , zero elsewhere, is an unbiased estimator of θ and has variance θ2/n.
Suppose you have a random sample {X1, X2, X3} of size n = 3. Consider the following three possible estimators for the population mean u and variance o2 Дi 3D (X1+ X2+ X3)/3 Ti2X1/4 X2/2 X3/4 Дз — (Х+ X,+ X3)/4 (a) What is the bias associated with each estimator? (b) What is the variance associated with each estimator? (c) Does the fact that Var(i3) < Var(1) contradict the statement that X is the minimum variance unbiased estimator? Why or...
Exercise: Let Yİ,Y2, ,, be a random sample from a Gamma distribution with parameters and β. Assume α > 0 is known. a. Find the Maximum Likelihood Estimator for β. b. Show that the MLE is consistent for β. c. Find a sufficient statistic for β. d. Find a minimum variance unbiased estimator of β. e. Find a uniformly most powerful test for HO : β-2 vs. HA : β > 2. (Assume P(Type!Error)- 0.05, n 10 and a -...