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.

(a) Show that P(k = 0) = e(^+) and P(k = 1) = ( + )e(^+). 

(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 → ex to show that P(k = 0) ≌ e-λ and P(k = I) ≌  λe-λ . Extend this result to show that k is such that 

p(k)= exp(-2) 2k k!

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

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