Exponential Distribution
The exponential distribution is the probability distribution of the time between events in a Poisson point process with rate .
The mean of the exponential distribution is , the median of the exponential distribution is
, and the variance of the exponential distribution is
.
The PDF of the exponential distribution is .
The CDF of the exponential distribution is .
The inverse CDF of the exponential distribution is .
The minimum of exponential random variables with rates
is an exponential random variable with rate
, and the probability that exponential random variable
achieved that minimum is
.
The minimum of identically-distributed exponential random variables with rate
is an exponential random variable with rate
, and the probability that exponential random variable
achieved that minimum is
.
Normally, the exponential distribution can have values in the range . What if we wanted to truncate the distribution at
, i.e., sample values from an exponential distribution with rate
but disregarding samples outside of the range
for some constant
? The CDF of the truncated exponential distribution is the following:
The inverse CDF of the truncated exponential distribution is the following: