3.5 Student's t distribution  3. Probability Distribution Functions  3.3 The binomial distribution

3.4 Poisson distribution

The Poisson distribution applies to the number of randomly occurring and countable events that occur in an interval. For example, the expected average rate R at which cloud droplets are detected by an airborne counter is given by

R = A V c  (3.6)

 where A is the sample area within which passing droplets are counted, V the airspeed, and c the droplet concentration. However, the droplets counted during any particular interval will differ from R because only discrete and not fractional events can be counted, and because statistical fluctuations will cause the number counted to vary from the true population mean. The binomial distribution appears to be applicable, because it gives the probability of seeing a given number if the probability p is known. However, in this case, the number of locations at which a droplet could be observed is infinitely large, and the probability of an observation at each location infinitesimally small. The appropriate distribution is therefore the limit of the binomial distribution as the number of possible events approaches infinity while the probability of any specific event approaches zero, maintaining the correct average number of events in a specific interval. This limit is the Poisson distribution function.

The Poisson distribution function thus gives the probability of observing n discrete events if the true mean is $\mu$. It has the form

$$\Phi_P(n;\mu) = {{\mu^n}\over{n!}} e^{-\mu} .$$ (3.7)

 

The mean of this distribution is $\mu$, and the variance is also $\mu$. This is the source of the common estimate that the standard deviation in a counted number of events is the square root of the number counted. Figure 3.4 shows that the distribution function, even for small numbers of events, is similar to a Gaussian distribution.

 

Figure 3.4: Poisson distribution functions (3.7), shown as heavier lines, for cases having means 6 and 12. The Gaussian distribution functions having the same means and standard deviations are also shown as thinner smooth lines. Values for the Poisson distribution function are plotted only at integer values, and adjacent values are connected by straight lines.

   
 3.5 Student's t distribution  3. Probability Distribution Functions  3.3 The binomial distribution