Statistical Sampling

Exponential distribution

Assume that an event occurs at irregular, random intervals. Then the time T between two consecutive events follows an exponential distribution. Assuming that the event happens at a frequency \lambda, the probability that a second event occurs at the latest a time t after a first one has occurred, is P(T \le t) = 1 - e^{-\lambda t}, which approaches 1 with increasing t. The smaller the value of the frequency \lambda, the slower it approaches one, i.e. the longer time may pass between consecutive events.

The typical example cited is that of radioactive decay. When an atomic nucleus in a piece of radioactive matter decays, a particle is emitted which can then be registered by, for example, a Geiger counter. The frequency with which this occurs depends on the type of nuclues, that is, what element it is, as well as what isotope. Since this frequency of events characterizes the element, one may thus talk about the expected time it takes for, say, half of the nuclei in a piece of a given element to have decayed. Hence the term 'half-life'.

Basic properties:
Notation:
X \sim \mathrm{Exp}(\lambda)
Type:
Continuous
Parameters:
\lambda \in \mathbb{R}_{> 0} = (0, \infty)
Variables:
X
Support:
\mathbb{R}_{\ge 0} = [0, \infty)
PDF:
f(x) = \lambda e^{-\lambda x}
CDF:
F(x) = P(X \le x) = 1 - e^{-\lambda x}
Mean:
\frac{1}{\lambda}
Variance:
\frac{1}{\lambda^2}