A variable x is Poisson-distributed if it obeys the equation
| $P(x) = \dfrac{\lambda^xe^{-\lambda}}{x!}$ | (1) |
where $\lambda$ is the mean of the distribution. The Poisson distribution is a discrete distribution for integer values of $x$.
Poisson distribution for events with rate r
We usually encounter the Poisson distribution in this course when considering events that occur with an average rate r. Suppose we observe such a system for time t and count how many events we observe. The mean value of events we expect is $\lambda = rt$, however we may sometimes observe more or less. The distribution is then
| $P(n,rt) = \dfrac{(rt)^ne^{-rt}}{n!}$ | (2) |
where, again, $n$ is a discrete variable and $rt$ may be a continuous variable. The Poisson distributions for fixed values of $n$ as a function of scaled time $rt$ are show in Fig 1.
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| Figure 1: Poisson distribution as a function of mean time for fixed values of n. |
We see that each distribution is peaked around the mean time $\lambda = rt$, and that the probabilities decay to zero for long time. A few special cases are worth pointing out.
- In the case of $n=0$ (that is, the case where no event occurs in the time interval), the probability is a pure exponential.
- In the case of $n$ large, the Poisson distribution approaches the Gaussian (Normal) distribution with mean $\lambda$ and standard deviation $\sqrt\lambda$:
| $P(n,\lambda) = \dfrac{1}{\sqrt{2\pi\lambda}}e^{\frac{-(n-\lambda)^2}{2\lambda}}$ | (3) |
Uncertainty in count
Let us consider the common situation of measuring the decay rate of a radioactive element.
In a typical experiment, we may collect $N$ counts (e.g. the count on an electronic scaler or singles detector, or the number of counts in a single PHA channel or region of interest) during a measurement of time $t$. The exact value of $N$ that we get will be random, but we expect that the likelihood of any given value of $N$ is given by a Poisson distribution with (true) mean $N_{avg} = rt$ (where $r$ is the true average rate). Our best estimate (after one measurement) of the true mean is therefore $N$ and our uncertainty is $\sqrt{N}$ .