Question

Prove that for a sample of n where X_i ~ iid Bernoulli (p) and Y_n = ∑X_i, that

Bn =                [Yn – np] / √[np(1-p)]  -->D N(0,1)

i.e. B_n has a limiting distribution of the standard normal. No need to use the MGF, you can use a theorem to answer this (which you must identify). Show all steps and parameterize the RV as the theorem specifies

Answer 1

This can we proved by using central limit theorem.

Let be a set of independent random variates and each have an arbitrary probability distribution with mean and a finite variance . Then the normal form variate

has a limiting cumulative distribution function which approaches a normal distribution.

Here X follows Bernoulli distribution . Hence its mean and variance are

and

Put all these values in above equation,

We see than Bn is same as z. Hence Bn has a limiting cumulative distribution function which approaches a normal distribution.