Exponential Distribution

  The exponential distribution is characterized by a probability density function

with positive a and for , resulting in

Exponential distributions describe the distance between events with uniform distribution in time: if x is the time variable, ax is the expected number of events in the interval [0,x], then is the probability of no event in [0,x] ( Poisson Distribution). The probability for the first event to occur in the interval is given by

Thus, the distribution of individual lifetimes of unstable particles is exponential Exponential functions are also commonplace when describing phenomena of attenuation. Depending on the context, the mean 1/a is called the mean life of a particle, the lifetime of a stored beam, the attenuation length of a scintillator, etc.

In a bin of width with starting abscissa x₁ one will find a fraction of events given by

where . The average height for the bin is given by .

The average abscissa for the same bin is at

which is always between x₁ and as can be seen from the development