f(X; , ) = 1 ()X1eX/ for X >0 = 0 elsewhere where and > 0 and () = ( 1)! This

f(X; α, β) = 1 Γ(α)βαXα−1e−X/β for X >0 = 0 elsewhere where α and β > 0 and Γ(α) = (α − 1)! This is a skewed and continuous distribution.

(a) Show that E(X) = αβ and var(X) = αβ2.

(b) For a random sample drawn from this Gamma density, what are the method of moments estimators of α and β?

(c) Verify that for α = 1 and β = θ, the Gamma probability density function reverts to the Exponential p.d.f. considered in problem 9.

(d) We state without proof that for α = r/2 and β = 2, this Gamma density reduces to a χ2 distribution with r degrees of freedom, denoted by χ2r . Show that E(χ2r ) = r and var(χ2r ) = 2r.

(e) For a random sample from the χ2r distribution, show that (X1X2, . . .,Xn) is a sufficient statistic for r.

(f) One can show that the square of a N(0, 1) random variable is a χ2 random variable with 1 degree of freedom, see the Appendix to the chapter. Also, one can show that the sum of independent χ2’s is a χ2 random variable with degrees of freedom equal the sum of the corresponding degrees of freedom of the individual χ2’s, see problem 15. This will prove useful for testing later on. Using these results, verify that the sum of squares of m independent N(0, 1) random variables is a χ2 with m degrees of freedom.