1) Suppose X1,X2,...,Xn are iid Gamma(α,λ) random variables. a) Express the ﬁrst and third moments (µ1...

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.

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orchestra answered 1 year ago

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