# Suppose that m and n have independent Poisson distributions of means and , respectively (see question

## Question:

Suppose that m and n have independent Poisson distributions of means λ and µ, respectively (see question 6) and that k = m + n.

(b) Generalize by showing that k has a Poisson distribution of mean λ + µ.

(c) Show that conditional on k, the distribution of m is binomial of index k and parameter λ / (λ+ µ,).

**Question 6.**

Suppose that k ~ B(n,π) where n is large and π is small but n π = λ has an intermediate value. Use the exponential limit (1 + x )^{n} → e^{x} to show that P(k = 0) ≌ e^{-}^{λ }and P(k = I) ≌ λe^{-}^{λ }. Extend this result to show that k is such that

that is, k is approximately distributed as a Poisson variable of mean λ.

Fantastic news! We've Found the answer you've been seeking!

## Step by Step Answer:

**Related Book For**