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The exponential distribution is a special case of the gamma distribution. It is a continuous distribution that is used to describe the probability distribution of the time elapsed between events in a Poisson process. This refers to those processes in which events occur continuously and independently of each other, but at a constant average frequency.
The exponential distribution follows the following probability function:
where X is a continuous random variable and lambda ( λ ) is a characteristic parameter of each particular distribution. The following figure shows the graph of this distribution function for different values of λ.
As can be seen, this function decays exponentially from an initial value equal to λ and approaches zero asymptotically as x increases.
The mean of this distribution function is given by μ = 1/ λ and its variance is σ 2 = (1/ λ) 2 . The following sections show how to calculate the median.
Importance of the exponential distribution
As mentioned at the beginning, the exponential distribution can be applied to any system that follows a Poisson process. This means that it serves to describe the times between events such as customer arrivals at service facilities, the times between failures of electronic systems or components, and the survival of living beings.
What is the median?
Before we proceed to calculate the median, we need to understand what it is. The median of a probability distribution corresponds to the value of the random variable that divides the distribution in half. In the case of discrete variables, this means leaving the same number of values on both sides of the median. For the exponential function and the other continuous distribution functions, the median is the point that leaves the same area under the probability density curve on both sides.
Another more practical way of looking at the median, and which is the one we will use to find it in this article, is that it corresponds to the point at which the distribution function has a value of 0.5. That is, it corresponds to the solution of the following equation:
Calculation of the median of the exponential distribution
To find the median of the exponential distribution, we will use the distribution function and find the value of the random variable for which the distribution function has a value of 0.5, as explained in the previous section. In other words, we will say that the median (Me) is the value of the random variable, x, for which it is verified that:
All we need to do now is plug in the pdf ( f(x) ) corresponding to the exponential distribution and integrate:
Where we have made use of the piecewise definition of the probability distribution function, which has a value of zero for all values of the random variable less than or equal to zero. This is a simple integral:
Now, we set equal to ½, and we solve the equation to find the median, Me.
Finally, it is rearranged, the natural logarithm is taken on both members and Me is cleared:
Therefore, the median of the exponential distribution is given by ln2/λ.
The bias of the exponential distribution
If we compare the value of the median that we just obtained, ln2/λ, with the value of the median of this distribution that we mentioned at the beginning, 1/λ, we quickly realize that the median is less than the mean, because ln2 is a number less than 1.
Whenever the mean does not coincide with the median, the distribution is said to be skewed. Since in this case the mean is greater than the median, the exponential function is said to be skewed to the right .
Because the median is a measure of central tendency that is less sensitive to extreme values than the mean, in cases like this where bias is determined to exist, it is preferred to use the median to represent that central tendency.
References
LesKanaris. (nd). How to calculate the median of the exponential distribution – Interesting – 2021. Retrieved from https://us.leskanaris.com/2916-exponential-distribution-medians.html
Lifehackk. (2018). How to calculate the median of the exponential distribution – 2021. Retrieved from https://esp.lifehackk.com/14-calculate-the-median-of-exponential-distribution-3126442-7366
Simple Mathematics. (2021, September 6). Median – exponential distribution [Video file]. Recovered from https://www.youtube.com/watch?v=0s3h1Tfysog
Mtz De Lejarza E., J., & Mtz De Lejarza E., I. (1999). Exponential distribution. Retrieved from https://www.uv.es/ceaces/base/modelos%20de%20probabilidad/exponencial.htm
