Question

Use the Central Limit Theorem to show

i. Po(λ) → N(µ, µ) for large µ.

ii. Bin(n, π) → N(nπ, nπ(1 − π)) for large n, provided neither nπ nor nπ(1 − π) is too small

Answer 1

The central limit states that the sample mean will be approximately normally distributed for large sample sizes regardless of the distribution from where we are sampling.

i) Let Y be a Poisson variable with rate . The Y can be the number of occurrences in the experiment that runs for time - that is same as if there are experiments that each run independently for time 1 unit. Hence we can add their result .

Again Y is a sum of n independent identically distributed random variable. Hence central limit theorem states that , for large ,Y has an approximate normal distribution with

Thus Po(λ) → N(µ, µ) for large µ. Where µ is same as .

ii.) Let X be a variable following Bin(n, π)

X is the result of repeating the same Bernoulli experiment n times and looking at the total number of successes. We can write X as the sum of n Bin(1, π) variable .

i.e.

X is the sum of n independent identically distributed random variables. Then central limit states that, for large n i.e. large number of Bernoulli experiments, X will have an approximate normal distribution

NOTE: Expectation of sum of random variable is equal to sum of expectations only when the random variables are independent identically distributed. Similar arguments can be given for writing the formula of variance in the above 2 derivations.