a. Refer to Problem 14.6. If T_n is Poisson, show √T_n has asymptotic variance 1/4.

b. For a binomial sample with n trials and sample proportion p, show the asymptotic variance of sin^-1(√p) is 1/4n. [This transformation and the one in part (a) are variance stabilizing, producing variates with asymptotic variances that are the same for all values of the parameter. Traditionally, these transformations were employed to make ordinary least squares applicable to count data.]

Data from Problem 14.6:

Suppose that T_n has a Poisson distribution with mean λ = nµ, for fixed µ > 0. For large n, show that the distribution of log T_n is approximately normal with mean log(λ) and variance λ^–1. [By the central limit theorem, T_n/n is approximately N(µ, µ/n) for large n.]