2. Let X1,X2, . . . ,Xn be a random sample from G(,), >0, >0: (a) Show that 4 = 3(+2)/4. (b) Show that

2. Let X1,X2, . . . ,Xn be a random sample from G(α,β), α>0, β >0:

(a) Show that

μ4 = 3α(α+2)/β4.

(b) Show that var

(n−1)

S2

σ2

≈ (n−1)



2+

6

α



.

(c) Show that the large sample distribution of (n−1)S2/σ2 is normal.

(d) Compare the large-sample test of H0 : σ = σ0 based on the asymptotic normality of (n−1)S2/σ2 with the large-sample test based on the same statistic when the observations are taken from a normal population. In particular, take α = 2.