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Question: suppose that x1 xn are a...

Question details

Suppose that x1, . . . , xn are a random sample from a B(α, β) distribution:

f(x; α, β) =  \frac{\Gamma (\alpha +\beta )}{\Gamma (\alpha) + \Gamma (\beta )}  x^(α-1) (1-x)^(β-1)

Here E[X] = α/(α + β) and E[X^2 ] = ((α + 1)α)/{(α + β + 1)(α + β)}.

(a) Show that the method of moments, using the first two moments, gives the equations

0 = α(1 − m1 ) − βm1

m1 − m2 = α(m2 − m1 ) + βm2

(b) Determine the method of moments estimator of α and β based on the first two moments.

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