Poisson Approximation to Binomial Distribution For a binomial d... | Filo

Normal Approximation to the Binomial

Prove the Poisson approximation of binomial probabilities.Claim: If X   Bin(n, p) with np = λ (or p = λ/n) and n is large, then P(X = i) can be approximated by the Poisson probability with rate λ.Hint: Use (1 - λ/n)^i, (1 - λ/n)^n-i, and (n - i 1)^n as n → ∞. The last assumption implies that we use the approximation for moderate i only.Which of the following arguments justifies the Poisson approximation? All limits are as n → ∞.(a) lim (np) = λ, by assumption(b) lim [(1 i/n)^(-