The exponential distribution can be used to determine the probability that it will take a given number of trials to arrive at the first success in a Poisson distribution; i.e. it describes the inter-arrival times in a Poisson process. It is the continuous counterpart to the geometric distribution, and it too is memoryless.
Definition 1: The exponential distribution has probability density function (pdf) given by
Excel Function: Excel provides the following function for the exponential distribution:
EXPONDIST(x, λ, cum) where λ is the parameter in Definition 1 and cum = TRUE or FALSE
EXPONDIST(x, λ, FALSE) = f(x) where f is the pdf value at x as defined above
EXPONDIST(x, λ, TRUE) = F(x) where F is the cumulative distribution function value at x corresponding to f above
In Excel 2010, 2013 and 2016 there is the additional function EXPON.DIST which is equivalent to EXPONDIST.
Observation: The exponential distribution is equivalent to the gamma distribution with α = 1 and β = 1/λ. Thus, EXPONDIST(x, λ, cum) = GAMMADIST(x, 1, 1/λ, cum).
There is no EXPON.INV(p, λ) function in Excel, but GAMMA.INV(p, 1, 1/λ) can be used instead.
The cumulative distribution function is
Other key statistical properties are:
- Mean = 1 / λ
- Median = ln 2/λ
- Mode = 0
- Range = [0, ∞)
- Variance = 1 / λ2
- Skewness = 2
- Kurtosis = 6