B4. Let X1, X2, ..., X, be a random sample from a normal distribution with known meanµ, unknown variance o? and density function1fx(x)IER.V2no2Using the log-likelihood, find the maximum likelihood estimator for o?. If the mean isµ = 3 and the sample is {r1, 2, T3} = {-1.2,0.5, 1.5}, what is the estimate for o??

Given that X1, X2, ..., Xn be a random sample from a normal distribution with known mean μ, unknown…

Answered: B4. Let X1, X2, ..., X, be a 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, ..., X, be a random sample from a normal distribution with mean µ and variance o?. (a) Find an unbiased estimator for o?. ( b) Show that Σ(x-Χ? Σ (Χ )- nX. i=1 i=1 (c) Find an estimator for o? using the method of moments. Is this estimator bi- ased/unbiased? Is it consistent?
Q2: Let X and Y have a bivariate normal distribution with parameters 4, = l2 = 1, of = 16, o = 25 and p = 1. The value of E (x\y) is
Q. 44 If f(x)=ce (1²-61+9)/32 , -∞
Indicate whether each of the statements below is True or False. No explanation is needed. i If random variable X has a larger expected value than random variable Y then it must also have a larger variance. ii Let random variable X be the height of a random CS5002 student in feet and X' be the height of a student from the same group in millimeters?, The variance of X' is greater than the variance of X. iii For every positive a, b eR there exists an no such that: n > no → an³ > bn² 'Unless you have made arrangements with the Professor prior to the start of this exam 2 Assume that not all students have the same height 1 Use the function definitions immediately below for the next 3 problems: f(n) = 3n log10 n + 5 g(n) = 99999 log3 n h(n) = .00001 logą n+.0000001n? iv f(n) = 0(h(n)) v g(n) = 0(h(n)) vi f(n) = 2(g(n)) vii An undirected graph on n nodes with at least ()/2 edges must be connected.
Let X and S? be the sample mean and sample variance calculated from nj number of Normal (u1, 01) random variables. Let X, and S; be the sample mean and sample variance calculated from n2 number of Normal (2, 02) random variables. (a). Show that X - X, is an unbiased estimator of u1 - µ2. (b). Find the standard deviation of X1 - X2. (C). Propose an estimator to estimate the standard deviation of X - X2. (d). Suppose that the two samples have the same variance o? = o; = o?. Show that S3 = (n -1)S+(n2-1)S n1+n2-2 is an unbiased estimator of o?. (e). In the setting of part (d), compute the relative efficiency of S? to S for estimating o2.
2. Let X be a rs from a distribution with known variance o² and unknown mean 8. Find a (1a) CI for 0. Assume n is large.
Q.7 For an exponential distribution fx(x; 4) = {4e*; Sue Hx, x 2 0 e.w. where u > 0 Write (Don't derive) (a) Mean (expected value), i.e. EX). (b) Standard deviation, i.e. S. D. (X). (c) Moment generating function, i.e. Mx(t). (d) Cumulative distribution function (cdf), Fx(x). of Nizy
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Suppose that X1,..., X, is a random sample from a normal distribution with mean u and variance o?. Two unbiased estimators of o? are ởf = s° = E(x, - Xy°, and ô = (X1 - 2 Find the efficiency of ở relative to ôž.
* Let X1, X2, ..., X, be an independent sam- ple from a normal distribution with unknown mean u and variance o?. Show that the pair (X,5), where X=-Ex,. 5 =(x, - X)'. i=1 is a sufficient statistic for (u, o²). Given A> 0, consider AS as an estimator of o². For what values of A is AS (i) (ii) unbiased? maximum likelihood, Which value of A minimises the mean square error E(A5³ – o*)?
1. Let X have the pdf f(x) = B-lexp{-x/B}, 0 < x
Suppose you take independent random samples from populations with means μ1 and μ2. Furthermore, assume either that (i) both populations have normal distributions, or (ii) the sample sizes (n1 and n2) are large. If X1 and X2 are the random sample means, and S1 and S2 are the random sample standard deviations, then how does the quantity behave? Give the name of the distribution and any parameters needed to describe it.

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