Poisson Distribution

The Poisson distribution applies in cases in which the probability for getting $A$ is very small compared with other possible outcomes. In that case we would use the binomial formula with a value of $k$ much smaller than $N$. For example, suppose we were counting radioactive decays as a function of time and we observe the decays over a time interval $dt$ that is much smaller than the decay lifetime, so the amount of radioactive material available for decay does not change noticeably during the time of observation. If the decay rate is $\lambda$ and we consider just one single atom, the probability that it decays in a time interval $dt$ is $p = \lambda dt$ (true as long as this is very small). Call this event $A$. If it doesn't decay (probability $1-p$) we call it event $B$. If we now consider $N$ atoms we can use the binomial distribution to give us the probability that that $k$ atoms out of $N$ atoms decay in the time interval $dt$. We expect that on average there will be $\bar k = pN$ decays. Let's find the probability for getting $k$ events in the limit of large $N$, if the expected (average) number $\bar k$ is constant as we take the limit. Notice that to keep $\bar k$ constant we have to decrease $p$ as we increase $N$. This means we are decreasing the time interval $dt$ as we increase $N$. To get the probability, we start with the binomial distribution, substitute $p = \bar k/N$ and take the limit

$$P(k,\bar k) = \lim_{N \rightarrow \infty}\frac{N!}{k!(N-k)!}(\bar k/N)^{k}(1-\bar k/ N)^{N-k}$$ (3)

After some algebra, using the Stirling approximation for the factorial and the Taylor expansion for the exponential function, we get the Poisson distribution:

$$P(k,\bar k) = \frac{\bar k^{k}e^{-\bar k}}{k!} .$$ (4)

This distribution is normalized to 1 as well. The sum generates the Taylor series for the exponential function:

$$\sum_{k=0}^{\infty} P(k,\bar k) = e^{\bar k}e^{-\bar k} = 1.$$ (5)

We will return to a discussion of properties of the Poisson distribution after discussion the Gaussian normal distribution.