Question

Let Xị, i = 1, ... be independent Bernoulli(p) random variables and let Yn = 1 1–1 Xį. (a) Use CLT to derive an approximate distribution for Yn. (b) Suppose that p + 1/2. Use Delta method to derive an approximate distribution for Yn (1 – Yn).

Answer 1

a)

E(Yn) = E(X) = p

sd(Yn) = sqrt(pq/n)

hence

Yn follow normal distribution with mean = p and variance = pq/n

b)

While the delta method generalizes easily to a multivariate setting, careful motivation of the technique is more easily demonstrated in univariate terms. Roughly, if there is a sequence of random variables X, satisfying V^{Xn – 6] “N(0,02), where and O2 are finite valued constants and denotes convergence in distribution, then n vñolg(X.) – 9(0) –> N(0,02 • [g'(O)]?) for any function g satisfying the property that g'() exists and is non-zero valued.

g(X) = x(1-x)

g'(x) = 1-2x

hence

Yn(1-Yn) follow N(p(1-p) , pq/n*(1-2p)^2)