# Exponential Distribution

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, λcum) 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) or –LN(1–p) or the following Real Statistics function can be used instead.

Real Statistics Function: The Real Statistics Resource Pack supplies the following function.

EXPON_INV(p, λ) = the inverse of the exponential distribution at p

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

How to do Damped Trend Linear Exponential Smoothing model using lynx dataset – Time Series Forecasting

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