Question

9. (Extra problem) (a) Explain why "Poisson(n) integer. nxNormal (1, 1/n)" if n is a large positive (b) Stirling's formula for approximation of factorials is: n!2 Use (a) to give a quick heuristic derivation of Stirling's formula by using a Normal approximation in the calculation of Pr(X =n) = P(n - < X <n+), where X: Poisson(n). Hint: (x)dr f(0) x 2a for small a and any function /

Answer 1

a)

mean of poisson = $\lambda$

variance of poisson = $\lambda$

when n is large

Poisson (n) = N(n,n)   by CLT

= N(n*1 , n^2* 1/n)

= n N(1,1/n)      { Var(aX) = a^2 Var(X) and E(aX) = aE(X)}

b)

P(X = n) = e^(-n) n^n/n!

using normal approximation to calulate

P(n-1/2 < X < n+1/2)

= P(-1/2 < X -n < 1/2)

= P(-1/2/sqrt(n) < Z < 1/2/sqt(n))

= P(|Z| < 1/(2* sqrt(n))

= f(0) * 2/(2* sqrt(n))  

=1/sqrt(2 $\pi$ ) 1/sqrt(n)

hence

$\frac{1}{\sqrt{2 \pi n}} = e^{-n} \frac{n^n}{n!}$

hence

(a) 1tc = it

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