If the events follow a Poisson process with rate
g2019
, the times between two consecutive events,
g1846
, has an
Exponential distribution.
This can be shown using Equation (1), the probability mass function for the
Poisson distribution.
If no events occur in the fixed time period [
0,g1872
], then this is equivalent to writing
that the time until the next event,
g1846
, is greater than
g1872
:
g1842g4666g1846 > g1872g4667 = g1842g4666g1866g1867 g1867g1855g1855g1873g1870g1870g1857g1866g1855g1857g1871 g1867g1858 g1872ℎg1857 g1857g1874g1857g1866g1872 g1861g1866 g1872g1861g1865g1857 g1872g4667 = g1842g4666g1850 = 0g4667 = g1857
g2879g3090g3047
Therefore,
g1842g4666g1846 ≤ g1872g4667 = 1 − g1842g4666g1846 > g1872g4667 = 1 − g1857
g2879g3090g3047
,
which is the cumulative distribution function of the
Exponential distribution.
The probability density function for the Exponential distribution is
g1858g4666g1872g4667 = g2019g1857
g2879g3090g3047
,
g1872 > 0
The mean of the inter-arrival time of the events is
g1831g4666g1846g4667 =
g2869
g3090
.
The variance is
g1848g4666g1846g4667 =
g2869
g3090
g3118
.
The Exponential cumulative distribution functions (CDF) for different values of
g2019
are shown below. Recall
g1832g4666g1872g4667 = g1842g4666g1846 ≤ g1872g4667 = 1 − g1857
g2879g3090g3047
.