Suppose that X_1,..., X_n is a random sample from an Exponential distribution wi

Question

Suppose that X_1,..., X_n is a random sample from an Exponential distribution with scale parameter theta. Then the probability density of X, is given by f_X, (x; theta) = 1/theta e^- 1(0, infinity) (x). Find E[X_i]. Find the maximum likelihood estimator of theta. Be sure to verify that your estimator maximizes the likelihood. Determine whether or not the maximum likelihood estimator of theta is an unbiased estimator of theta. Find the distribution of the maximum likelihood estimator of theta using the moment-generating function technique. Using the result of part d). find a pivot quantity and use it to construct a 95% confidence interval for theta. You may assume that n = 10 for this part, and you may divide the tail probabilities equally in the two sides of the density of the maximum likelihood estimator.

Explanation / Answer

(a) E(x) = x p(x) = x.1/ . e-x/ I(0,) (x)    = 1/x.e-x/   1/x!, where it is assumed that I(0,) (x) = 1/x!

So E(x) = 1/* e-1/ e-(x-1)/   1/(x-1)! = 1/* e-1/ * e1/ = 1/

(b) Let x1,x2,x3, be a random sample of size n from the given population.

Then L = x.e-x/

Log L = log x - x/

Taking partial derivative we can find the maximiumlikelihood estimate.

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