# Exponential Distribution definition, formula with applications

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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/λ2
• 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 β1= 4.
• Measure of kurtosis, β2= 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