Identifying a Poisson distribution

Question

Suppose I have a customer arriving at a store according to a Poisson process with rate $\lambda$ customers per minute. I observe there are 18 customers arriving in the first half hour.

Question)

Let X be the number of customers to arrive in a period of 30 minutes. If customers arrive according to a Poisson process with rate $\lambda$ customers per minute, state the distribution of X, with papameters.

My thoughts:

Let X = # customers to arrive in a period of 30 minutes.

Let $\lambda$ = customers arrive in 1 min period.

Therefore, X ~ Poisson ($\lambda$) or is it X ~ Poisson($\lambda$/30)?

Can anyone please help me if I am going to the correct direction?

Answer 1

If the number of arrivals in $1$ minute has Poisson distribution with parameter $\lambda$, then the number $W$ arriving in a $t$ minute interval has Poisson distribution with parameter $\lambda t$.

In our case, $t=30$, so $X$ has Poisson distribution with parameter $30\lambda$.

One way to remember this is that the parameter of a Poisson is the mean. If the mean number of arrivals in a minute is $\lambda$, then the mean number of arrivals in $30$ minutes is $30\lambda$,

Remark: The specific question that you asked, and that was answered above, may not be the question you meant to ask. In the preamble, it said that $18$ customers arrived in $30$ minutes. Then $18$ is a possibly quite poor estimate of the mean number of arrivals in a $30$ minute period. If the Poisson is a reasonable model, and we close our eyes and assume that the mean number of arrivals in $30$ minutes is $18$, then the number of arrivals in a $1$ minute interval has Poisson distribution with parameter $\frac{18}{30}$.