b) Let X₁, X₂, Xn, denote a random sample of size n from a distribution withprobability density function:f(x;0) = {0(1-x)-(¹+0), x>0elsewhereDerive fisher's information of 0.

Given information: The p.d.f is given f(x,θ)=θ(1-x)-(1+θ) ,x>00…

Answered: b) Let X₁, X₂,..., Xn denote 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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b) Let X₁, X₂,..., X, denote a random sample of size n from a distribution with probability density function: i) ii) iii) f(x; 0) = {0(1-x)(¹+0), Find the maximum likelihood estimator of 0. Derive fisher's information of 0. suppose that 5 independent observations were taken from such a distribution: 4 1 3 5 4 and 0 = 0.7052 Find a 99% confidence interval for 0. x>0 elsewhere D
Let X₁, X₂, X3,..., Xn denote a random sample of size n from the population distributed with the following probability density function: ((0+1)xº, = {CO₂ + 0, f(x; 0) = Find the maximum likelihood estimator (MLE) of 0. if 0 < x < 1 elsewhere
7. Let X and Y be continuous random variables with joint probability density function given by fxy(xy): Sc, (0, 0 < x < y < 1 otherwise a) Find the value of c. b) Find the marginal distributions of X and Y. c) Find the conditional distribution of Y given X. d) Find the E(Y) and Var(Y).
Q1. The cumulative distribution function of the random variable X is F(x) = 0, (x+1)², 1- (1-2)², 1, for x 1 (a) Compute-P(0.5 < X <1)¶ (b) What is the probability density function of X.?
Let X1, X2, . X6 be an i.i.d. random sample where each X, is a continuous random variable with probability density function f(x) = e-(-0) , x > 0 Find the probability density function for X(6).
2.70 Let X and Y be random variables with joint density function (4xy, 0< x,y< 1, f(x,y)= {0, elsewhere. Find the expected value of Z = √X² + y².
Let X and Y random variables have independent Gamma distributions with X-Gamma(1, 6) and Y-Gamma(2, B). a. Find the joint probability density of Z, = X + Y, Z, = X+Y a. Find the marginal pdf of Z2.
Suppose that X and Y are independent and uniformly distributed random variables. Range for X is (−1, 1) and for Y is (0, 1). Define a new random variable U = XY, then find the probability density function of this new random variable.
Let X1, X2,...,X, be a random sample from a distribution with density function e if x > 0 f(x; 0) elsewhere What is the maximum likelihood estimator of 0 ?
b) Let Y,,Y2, .. , Yn denote a random sample from N(0,0) distribution with probability density function: f(y;8) = e V2n0 i) Show that f(y; 0) belongs to the 1-parameter exponential family. ii) What is the complete sufficient statistic for 0? Justify your answer. iii) Show whether or not, the maximum likelihood estimator is an unbiased estimator of 0. iv) Does the estimator attains the minimum variance unbiased estimator of 0.

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b) Let X₁, X₂,..., Xn denote a random sample of size n from a distribution with probability density function: f(x;0) = {0(1-x)(¹+6), x>0 elsewhere Derive fisher's information of 0.