# Exponential Distribution definition, formula with applications

### Exponential Distribution

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

$f\left&space;(&space;x;\lambda&space;\right&space;)=\lambda&space;e^{-\lambda&space;x};&space;x>&space;0,\lambda&space;>&space;0.$

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

• 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.