Non integer $\lambda$ in poisson distribuiton

Question

Here's the problem:

Admit that the number of phone calls received in a call center is a random variable that follows Poisson's distribution, with $\lambda$ = 30 per hour.

• Determine the probability that in the first 15 minutes the call center receives, at least, 2 calls.

What I tried: since $\lambda$ = 30/ hour $\rightarrow$ $\lambda$ = 0,5/ minute, but by the definition $\lambda$ can only be an integer, right?

Answer 1

There is no restriction that the $\lambda$ parameter of a Poisson distribution be an integer. It just represents the expected, i.e. the average, number of occurrences of an event in a given interval. Just like the average roll of a standard 6-sided die is 3.5, it's possible for the average number of calls-per-minute to be 0.5.

Answer 2

The PMF of a Poisson distribution $p(n;\lambda)$ defined by the expression $$p(n;\lambda) = \frac{\lambda^n }{n!}e^{-\lambda}$$ is a valid PMF for all $n\in \mathbb{N}$ and $\lambda >0$. There is no restriction that $\lambda$ must be an integer. This makes intuitive sense once we realize that $\lambda$ is the expected value of such an RV. So indeed 30 calls per hour on average translates to 0.5 calls per minute or 1 per every 2 minutes. You just need to use consistent units for your computations.