Let X1, ..., X, € [X], where X r.v. with pdf 0.00-110.1) (2) w.r.t. the unknown...
Let X1, ..., X, be a random sample with pdf defined as: f(x) = 2x exp{ –x?/0}, where > 0. The distribution of the MLE is: O None of the alternatives. o ên ~ Gamman,/n) Oô - Gammale, n/=) Oô - Exp(0) O 6 – Exp(9/1)
beta >0 74. Let X1, X2, ..., Xn be a random sample from the PDF 010105T10 by Disclado Ol betrov , a < x < oo, -o < a < oo, (a) Find the MLE of (a, b). (b) Find the MLE of Pa,p{X1 2 1}. guld brun onheilt f (x; a, B) = 8-1e--(x-a) gmax B>
Let X = (X1, X2) be a 2 x 1 random vector having joint pdf (1 x € (0, 1) ~ [0, 1] 10 otherwise. Find the probability P(X1 < 0.5, X2 < 0.5)
5. Let X1,...,Xn be a random sample from the pdf f(\) = 6x-2 where 0 <O<< 0. (a) Find the MLE of e. You need to justify it is a local maximum. (b) Find the method of moments estimator of 0.
Let pdf of a r.v. X be given by f(x) = 1, 0<x< 1. Find Elet).
3. Let X1,... ,Xn be a random sample from a population with pdf 0, otherwise, where θ > 0. (a) Find the method of moments estimator of θ. (b) Find the MLE θ of θ. (c) Find the pdf of θ in (b).
3. Let X1,... ,Xn be a random sample from a population with pdf 0, otherwise, where θ > 0. (a) Find the method of moments estimator of θ. (b) Find the MLE θ of θ. (c) Find the pdf of θ in (b).
6-x-4, 0x<2 0 1 2cych Exri If for two R.V. s X&Y the joint pdf is given by, otherwise Find Frix (o (1), Frix (alt), Ely/x-1]. var [Ylx-i] = E[^\x-]- (E[1\x=1])!
3. Let X X be a random sample from Uniform[0, where > 0 is unknown. (a) Show that = max{X,X is the MLE of 0. (b) Let the CDF of @ be F(-). Find F(t) for any t e R (c) Find the pdf of 0 Hint: Find the distribution function of Z maxX1,X. The first feu steps will be as follous: F2(2) P(Z) P (maxX, x) ) = P (XS2, X X,) Nert use the fact that Xis are...
Let X1, X2, ..., Xn be a random sample from the distribution with pdf f(3;6) = V porta exp ( 0) 10.02) for some parameter 2 > 0. (a) Find the MLE for 0. (b) Find the Cramér-Rao lower bound for the variance of all unbiased estimators of 0. (c) Find the asymptotic distribution of your MLE from part (a).