A geometric distribution has a pdf given by P(X=x) = p(1-p)^x, where x = 0, 1,...

A geometric distribution has a pdf given by P(X=x) = p(1-p)^x, where x = 0, 1, 2, ..., and 0 < p < 1. This form of the geometric starts at x = 0, not at x = 1. Given are the following properties:

E(X) = (1-p)/p, and Var(X) = (1-p)/p^2

A random sample of size n is drawn; the data are X1, X2, ..., Xn.

A. Derive the Fisher information function for the parameter p.

B. Find the Cramér-Rao lower bound (CRLB) for the variance of an unbiased estimator for p.

C. Find a sufficient statistic for the parameter p

D. Show that the sample mean, xbar, is an unbiased estimate of E(X) = (1-p)/p Find the variance of xbar.

E. Argue whether or not the sample mean is a minimum variance unbiased estimate (MVUE) of population mean, mew.

Expert Solution

**The last part is abusive.

orchestra answered 1 year ago

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