8. Let X, X be a random sample of size n = 2 from a population...

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8. Let X₁, X₂ be a random sample of size n = 2 from a population having gamma distribution with probability density function 1 - xα-1, f(x|a,0) = e x > 0, г(а) да where a > 0 and 0 > 0 are unknown parameters, and I'(a) is the usual gamma function. (a) Show that T = (X₁ + X₂, X₁X₂) is a minimal sufficient statistic; (b) Are T₁ = (ln(X₁X₂, X₁ + X₂) and T₂ = (In X₁ + ln X₂, ln(X₁ + X₂)) sufficient statistics; here, for any real number x, x| denotes the modulus of x; (c) Find the minimum variance unbiased estimator of V(X₁). 8. Let X₁, X₂ be a random sample of size n = 2 from a population having gamma distribution with probability density function 1 - xα-1, f(x|a,0) = e x > 0, г(а) да where a > 0 and 0 > 0 are unknown parameters, and I'(a) is the usual gamma function. (a) Show that T = (X₁ + X₂, X₁X₂) is a minimal sufficient statistic; (b) Are T₁ = (ln(X₁X₂, X₁ + X₂) and T₂ = (In X₁ + ln X₂, ln(X₁ + X₂)) sufficient statistics; here, for any real number x, x| denotes the modulus of x; (c) Find the minimum variance unbiased estimator of V(X₁).

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