1. Review on likelihood. Assume that Y₁ Y2 and each Y; is identically distributed and both Yi U(0,0) have the uniform distributed on interval [0,0], when 0 > 0. ~ a) Suppose that the observations yı = 1.348 and Y2 = 0.812 were obtained. De- termine the likelihood function corresponding to the data. b) What is the maximum likelihood estimate of the parameter 0? (Note. in order to obtain a nice solution, you should assume that the density function of uniform distribution has non-zero values on closed interval [0,0]).

Q3. Suppose that X;i, i = 1,2, .,n have an Exponential distribution with parameter 2. a) Find maximum likelihood estimator (Â) for 2. b) Is â is an unbiased estimator?

Let X₁, X2, X3, X4 be a random sample of size 4 from a population with the following distribution function f(x; 0) = xº e Г(7)В7' if x > 0 elsewhere Where, ß > 0. a) Find the maximum likelihood estimator of the parameter ß. b) Is the maximum likelihood estimator of the parameter ß unbiased or not? c) Find the variance of the maximum likelihood estimator of the parameter ß. d) What is the fisher information in the sample of size n about the parameter ß. e) What is the Cramer-Rao lower bound for the variance of unbiased estimator of the parameter ß? f) Is this maximum likelihood estimator an efficient estimator of ß?

Vo)) 4663%9:39 PM You Just now (2) The two basic SDEs. Solve the following SDEs and find the expectation and variance of X₁. (i) dX, X,dt - 2XdB₁, Xo = 2. (ii) dx = -2X du +4dBu, 1 ≤u≤ 4, X₁ = 3.

Q3) Let X1,X2,. X, be a random sample from the truncated exponential distribution with pdf: f(x,0) = Please else (a) Derive the method of moment estimator of 0; (b) Derive the MLE of 0. (c) Are the MOME and the MLE unbiased estimators for 0? (d) Compare the MSE of the MOME and the MLE for 0. Which one is a better estimator for 0

Q11. Let X1,X 2,.,X, be a random sample from a Poisson distribution with parameter 2. (a) Find the Cramer-Rao lower bound for the variance of any unbiased estimator of 1. (b) Find the maximum likelihood estimator (m.l.e.) of å. (c) Is this m.l.e. unbiased and does its variance attain the Cramer-Rao lower bound?

3.Suppose that Xi,i =1,2,...,n have an Exponential distribution with parameter delta. a) Find maximum likelihood estimator for delta. b) is estimator is an unbiased estimator.