Poisson distribution

The Poisson distribution is a discrete probability distribution. It expresses the probability of a number of events occurring in a fixed period of time if these events occur with a known average rate, and are independent of the time since the last event.

The probability that there are exactly k occurrences (k being a non-negative integer, k = 0, 1, 2, ...) is

$$f(k;\lambda)=\frac{e^{-\lambda} \lambda^k}{k!}$$

  • e is the base of the natural logarithm (e = 2.71828...),
  • k! is the factorial of k,
  • λ is a positive real number, equal to the expected number of occurrences that occur during the given interval. For instance, if the events occur on average every 4 minutes, and you are interested in the number of events occurring in a 10 minute interval, you would use as model a Poisson distribution with λ = 10/4 = 2.5.

(From the Wikipedia).

The detected photons in a CCD camera or a photomultiplier tube follow a Poisson distribution, which is responsible for the Photon Noise and determines de Signal To Noise Ratio of the acquired image.