Let X1,…,Xn be i.i.d. Poisson random variables with parameter λ>0 and denote by X¯¯¯¯n their empirical average,

Question

X¯¯¯¯n=1n∑i=1nXi.

Find two sequences (an)n≥1 and (bn)n≥1 such that an(X¯¯¯¯n−bn) converges in distribution to a standard Gaussian random variable Z∼N(0,1) .

Answer 1

Secondly, express P(|Z|≤t) in terms of Φ(r)=P(Z≤r) for t>0 .

Write Phi(t) (with capital P) for Φ(t) .

Answer 2

an is sqrt(n)/sqrt(lambda)
bn is lambda