Let X₁, X2 be independent random variables with mean μ and variance o2, and a is a constant.We also know that μ = o. Suppose we have two estimators of μ:P1 = (X₁ + X₂)A₂ = X₁ + X₂a) Calculate the bias of each estimator.b) What is the variance of each estimator?c) Calculate the relative efficiency of 2 with respect to ₁. Determine the constant a such that fbecomes a more efficient estimator.

Given that X1,X2 be independent random variables with mean μ adn variance σ2. So, E(X1)=μ=E(X2) and…

Answered: Let X₁, X₂ be independent random…

MATLAB: An Introduction with Applications

6th Edition

ISBN:9781119256830

Author:Amos Gilat

Publisher:Amos Gilat

Chapter1: Starting With Matlab

Section: Chapter Questions

Problem 1P

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Let X1, X2 denote a random sample from a population having mean u = 10 and variance o? %3D 1. Consider the following estimators of u: X1+X2 2 = 2X1 – X2 Round all answers to 1 decimal place. 1. The bias of O is and the bias of O2 is 2. The variance of O, is and the variance of O2 is 3. The relative efficiency of O2 to O1 is
Let X1, X2, X3 be random variables each having a mean u and variance o. Further, Cov(X1, X2) = 2, Cov(X1.X3) = 3 and Cov(X2, X3) = 1 Define U = 2X1 – X2 + 4 X3 Solve for the mean and standard deviation of U.
If x, x2 ... Xn is a random sample from N (µ, o²) popula- tion, find súfficient estimator for µ and ơ.
4. Two independent random variables x₁ and x₂ are both uniformly y₁=X₁X2/(x₁+x2). Hint: distributed between 9 and 11. Find the pdf of introduce the dummy variable y2=X2 and use the approach of two functions of two variables.
Let Y1, Y2, Y3, Y4, and Y, be i.i.d. random variables from a population with mean µ and variance o². Further, let Y = (Y, +Y½ +Y3 +Y4+Y;) denote the average of these five random variables. Now consider a different estimator of µ called W w = }Y + }Y½ + }Y, + }Y, + Y, What is the variance of W? That is, calculate Var|W] Hint: Use the properties of expectations, variances, etc. 187 800 800 187
Given W = (aX + Y)2 where X and Y are zero-mean random variables with variance 02X =2, and variance 02Y = 12, respectively. Moreover, their correlation coefficient is r=- 0.4. 1. Find the expected value of W as function of a 2. Find the value of the parameter a that minimizes the mean value of W. 3. Find the minimum mean value.
Consider an estimator of μ: W= 1/6 Y1+1/16Y2+1/4Y3+1/8Y4+1/2Y5 This is an example of a weighted average of the Yi's. Show that W is also an unbiased estimator of μ. Find the variance of W.
2) Let G and H be two independent unbiased estimators of 0. Assume that the variance of G is two times the variance of H. Find the constants a and b so that aG + bH is an unbiased estimator with the smallest possible variance for such a linear combination.
Suppose X and Y are two random variables with E[X] = 1, Var (X) = 4, E[Y ] = -1, Var (Y) 4, and Cov (X, Y) = 1. Find the standard deviation of (X - Y). = (a) 2 (b) √2 (c) 6 (d) √6 (e) None of the above
Show that if t" (t + t) where t and ť are both most efficient estimators with variance v, then var (t") = v (1 + p).
[Joint PDFS will be covered in Week 7] Let the random variables X and Y be the portions of the time in a day that two alternative routes between Topkapi and Uskudar have congestion (X for Route 1 and Y for Route 2). The joint PDF is given by fxy(x.y) = 2x2 + y? where Osxys1. (b) Assume Z=X+Y and W= XY and traffic experts are interested in the expected values of Z and W. E(Z) = and E(W) =O (Simplify your answers. Do not convert fractions into decimals.) (c) Find the variances of X and Y as well as the covariance between them. v(X) =D VY) =D and Cov(X, Y) =D (Simplify your answers. Do not convert fractions into decimals.) (d) The variance of Z can be found as V(Z) =- (Simplify your answer. Do not convert fractions into decimals.)
Suppose that Y₁, Y₂, ..., Ym is a random sample of size m from Gamma (a = 3, B = 0), where 0 is not known. Check whether or not the maximum likelihood estimator Ô is a minimum variance unbiased estimator of the parameter 8.

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