In probabilistic statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process where events occur continuously and independently at a constant average rate. It is a special case of the gamma distribution.

### Definition

A continuous random variable X is said to have an exponential distribution with parameter θ if its probability density fumction is defined as,

where, θ is the only parameter of the distribution and θ>0. Exponential distribution is continuous probability distribution and it has memoryless property like geometric distribution.

### Characteristics of Exponential Distribution

There are some impotant characteristics as following,

- Exponential distribution has only one parameter ‘λ’.
- Mean of exponential distribution (variate) is 1/λ.
- Variance of exponential distribution (variate) is 1/λ
^{2}. - Moments of all order exists in exponential distribution.
- Characteristic function of exponential distribution is .
- Moment generating function of exponential distribution is .
- Median of exponential distribution is also 1/λ.
- The measure of skewness β
_{1}= 4. - Measure of kurtosis, β
_{2}= 6.These measures show that exponential distribution is positive skewed and leptokurtic. - If the value of λ=1; mean= variance, if λ<1; mean<variance and if λ>1; mean>variance.
- It also process the memoryless property just like geometric distribution.

### Applications

According to wikipedia, there are some applications as following,

- It occurs naturally when describing the lengths of the inter-arrival times in a homogeneous Poisson process.
- The exponential distribution may be viewed as a continuous counterpart of the geometric distribution, which describes the number of Bernoulli trials necessary for a
*discrete*process to change state. In contrast, the exponential distribution describes the time for a continuous process to change state. - In real-world scenarios, the assumption of a constant rate (or probability per unit time) is rarely satisfied. For example, the rate of incoming phone calls differs according to the time of day. But if we focus on a time interval during which the rate is roughly constant, such as from 2 to 4 p.m. during work days, the exponential distribution can be used as a good approximate model for the time until the next phone call arrives.

Similar caveats apply to the following examples which yield approximately exponentially distributed variables:

- The time until a radioactive particle decays, or the time between clicks of a Geiger counter
- The time it takes before your next telephone call
- The time until default (on payment to company debt holders) in reduced form credit risk modeling