# What is the physical meaning of the parameter of a Poisson distribution?

I have done a laboratory session at my university where I had to check that the disintegration of nucleii follows a Poisson distribution

$$P(n)=\frac{\lambda^n}{n!} e^{-\lambda},$$

where $P(n)$ is the probability of the disintegration of exactly $n$ nuclei for a given time interval.

I measured the data using a computer program and I've been analyzing the data using MatLab but I don't know if the results that I get make sense or not.

The activity of the test sample was $3$ disintegrations per second and when analyzing the data that I got manually I get that it fits via a Poisson distribution of $\lambda=3.45$ but when using the data that I got via the computer I get $\lambda \approx 900$. I believe that this is an error in my calculations.

This leaded me to wonder what is the exact meaning of $\lambda$ in that expression. I understand that it is the mean number of nuclei disintegrated in a given time, but, does that mean that it depends on the size of the sample? What is the meaning of this parameter?

If you have a Poisson distribution with a probability $$P(n)=\frac{\lambda^n}{n!}e^{-\lambda}$$ that there will be $n$ events per bin, then $\lambda$ is the mean number of events per bin. You can get this via a direct calculation, $$⟨n⟩ =\sum_{n=0}^\infty nP(n) =\sum_{n=0}^\infty n \frac{\lambda^n}{n!}e^{-\lambda} =\lambda e^{-\lambda}\sum_{n=1}^\infty \frac{\lambda^{n-1}}{(n-1)!} =\lambda,$$ or if you're deriving the Poisson form for $P(n)$ in your problem then you'll start with essentially that understanding for $\lambda$ and work your way up.