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## (Basic Statistics for Citizen Data Scientist)

# 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, 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) or –LN(1*–p*)*/λ* 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 /
*λ*^{2} - Skewness = 2
- Kurtosis = 6

## Statistics for Beginners in Excel – Exponential Distribution

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