**2. Poisson distribution**

Rare and random incidents related to a time of reference, an area of reference or similar can be described by the Poisson distribution. Examples are the number of surface defects in body part stamping in the automotive industry or the number of calls in a call centre within a given time.

If the probability of an incident per time is known to be *p*, then within the time interval *T*, we expect a total number of *λ* = *pT* incidents. But the actual number of incidents within *T* will fluctuate randomly. The Poisson distribution allows us to calculate the probability of a given number *x* of events within *T*:

$$\text{Probability of x includes if } \lambda \text{ includes are expected = } \frac{e^{-\lambda}\lambda^x}{x!} \tag{1}$$

If the time of reference, *T*, is 1 year, then *λ* is the expected number of incidents within 1 year. If *λ* is much smaller than one, then it is also the probability of one incident within 1 year and of at least one incident per year. Analysing not only one but many reactors, the expected total number of accidents is simply the sum of the expected number for each single reactor, and, as long as the reactor incidents are independent of each other, the actual number of accidents is Poisson distributed.

In analysing real systems, the number of (statistically fluctuating) incidents *x* is known, and *λ* has to be determined. Then, the best estimate for *λ* is simply this empirical value *x*. However, this estimate is not necessarily the true value of *λ* because the incidents occur randomly. Poisson statistics allow us to compute an interval that contains the true value of *λ* with a confidence level α (typically 90, 95 or 99%), the so-called confidence interval. This is determined by calculating two values, *λ*<sup>1</sup> and *λ*<sup>2</sup> , for a given number of incidents *x*. For the 95% confidence interval, we choose *λ*<sup>1</sup> <sup>&</sup>lt; *<sup>x</sup>* such that the probability of observing *x* or more events is 2.5% and *<sup>λ</sup>*<sup>2</sup> <sup>&</sup>gt; *<sup>x</sup>* such that the probability of observing *x* or fewer events is 2.5%. Then, the interval *λ*<sup>1</sup> to *λ*2 is a 95% confidence interval. This means that if we study many cases, then in 95% of these cases, the true value of *λ* lies within this interval. The more cases we observe the narrower the confidence interval will be and the closer the estimate of *λ* will be to the true value.

As an example, suppose that the empirical number of events is *x* = 4. Then, the Poisson distribution with a value for *λ* equal 1.090 gives the probability that the number of events is greater than or equal to 4 to be 2.5%. If *λ* is 10.242, then the probability that the number of events is less than or equal to 4 is also 2.5%. Thus, for the empirical value of *x* = 4, we say that the true value for *λ* is between 1.090 and 10.242 with 95% confidence.

A similar measure of the probable distance between the estimated empirical value and the true value is the standard error. In large samples the probability that the distance between the estimated and the true value is less than the standard error is approximately 68%.
