Problem 4: Let X1, X2,..., Xn be a random sample of size n from the uniformpopulation given by11 for 020 elsewhere.1f(x;0)Show that if Y1 and Yn are the first and nth order statistic, any estimatorsuch that11n22can serve as a maximum likelihood estimator of 0. What does this saythe uniqueness of the maximum likelihood estimatorabout

Let X1, X2 ……….Xn be a random sample of size n from the uniform population.Probability density…

Answered: Problem 4: Let X1, X2,..., Xn be a…

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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3. Let X1, X2, ..., Xn be a random sample from the population N(u, o²) and Y1, Y2,...,Ym a random sample from the population N(uy, o²), where both means (ua and uy) are assumed known. Note that the two distributions have a common variance of o?. (a) If the X's and Y's are independent, show that the MLE for the common variance is: E(Xi - Ha)² +E(Y; - Hy)² n + m (b) Is the MLE found above unbiased for o??
show that if X11, X12,..., X1, X21, X22,..., X2n₂ are independent random variables, with the first n constitut- ing a random sample from an infinite population with the mean μ₁ and the variance of and the other n2 constitut- ing a random sample from an infinite population with the mean μ2 and the variance o2, then 1 M2 (a) E(X₁-X₂) = μ₁ −μ2; 07 0 22 (b) var(X₁-X₂) = + n₁ n₂
Suppose that X, Y, and Z are jointly distributed random variables, that is, they are defined on the same sar Suppose that we also have the following. E(X)=-5 E(Y)=8 E(Z)=6 Var (X)=24 Var (Y) = 16 Var (Z) = 33 Compute the values of the expressions below. E (4Y+4)= 36 E(-4Z+2X)- -3 = 11.33 Var (4Y)-2= 254 E(-x²) = 49 0|0 X
Question 3. Let X₁,, Xn be a random sample from a distribution with the pdf given by f₁₁x(x) = exp(-ª) if x ≥ A, otherwise f(x) = 0, where > 0. Find the MLE's of 0 and X. Start by writing the likelihood function and note the constraint involving A. Question 4. #4.2.9 of the textbook.
3. If x1, X2, X3, ..., Xn is a random sample from a normal 1 distribution N (µ, 1) show that t= E xí is an unbiased estimator of n i = 1 H² + 1.
Let X11 X12, Xini and X21, X22, X2n2 be two independent random samples of size n₁ and n₂ from two normal populations N(₁, 2) and N(2, 2) respectively. (a) Derive the maximum likelihood estimators (mle's) of all the parameters in the first population (X₁). Using analogy, state the mle's of the parameters of the second population. (b) Find the pooled estimator of the common variance when it is assumed that of = = ². Suggest an unbiased estimator of o². |

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