Question

Please show your steps for the question below(include R codes if possible):

Let X1, X2, ..., X30 be i.i.d Poisson random variables with a mean of 5. Find the probability that the sample mean X is between 4 and 6.

Answer 1

this is Poisson distribution with parameter λ=5

here mean of distribution=μ=λ=			5
varaince =σ2=λ=		5
standard deviation σ=√λ=		2.236

fro n=30 and normal approximation:

here mean=       μ=	5
std deviation   =σ=	2.236
sample size       =n=	30
std error=σx̅=σ/√n=	0.40825

probability =P(4<X<6)=P((4-5)/0.408)<Z<(6-5)/0.408)=P(-2.45<Z<2.45)=0.9928-0.0072=0.9857

(R code:

> pnorm(6,5,sqrt(5/30))-pnorm(4,5,sqrt(5/30))
[1] 0.9856941