In: Statistics and Probability
1) Suppose X1,X2,...,Xn are iid Gamma(α,λ) random variables.
a) Express the first and third moments (µ1 = E(X) and µ3 = E(X3)) as functions of λ and α.
b) Solve this system of equations to find estimators of ˆ λ and ˆ α as functions of the empirical moments. Note that the estimates must be positive.
2) Suppose that X1,X2,...,Xn are a iid Beta(a, a) random
variables. That is, a beta distribution with the restriction that b
= a. Using the formulae for the expectation and/or variance, find a
method of moments estimator for a.