Let X1, X2..., Xn are i.i.d. random variables from a Binomial distribution with parameters m and p where p is the probability of success. The probability density function is:
$P(X = x|m,p) = \binom{m}{x} p^x (1 - p)^{m-x}$ $x = 0,1,.., m$ $0 < p < 1$
a) Obtain a method of moments estimator $\hat{p}_{MOM}$ for p. Calculate an estimate using this estimator when $x_1 = 3, x_2 = 6, x_3 = 4, x_4 = 3, x_5 = 6$, assuming $m = 5$.
b) Is this estimator ($\hat{p}_{MOM}$) unbiased?
c) Obtain a maximum likelihood estimator $\hat{p}_{MLE}$ for p. Calculate an estimate using this estimator when $x_1 = 3, x_2 = 6, x_3 = 4, x_4 = 3, x_5 = 6$, assuming $m = 5$, as well as $\hat{P}(X = 4)$ for $m = 8$.
d) Is this estimator ($\hat{p}_{MLE}$) unbiased?
e) Calculate the Fisher information function for Binomial distribution.
f) Using the Fisher information, calculate the Cramer Rao lower bound for the variance of any unbiased estimator of p.
g) Show that sample mean achieves the Cramer Rao lower bound.
