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Statistics/Distributions/Exponential

Exponential DistributionEdit

Exponential
Probability density function
 
Cumulative distribution function
 
Parameters λ > 0 rate, or inverse scale
Support x ∈ [0, ∞)
PDF λ e−λx
CDF 1 − e−λx
Mean λ−1
Median λ−1 ln 2
Mode 0
Variance λ−2
Skewness 2
Ex. kurtosis 6
Entropy 1 − ln(λ)
MGF  
CF  

Exponential distribution refers to a statistical distribution used to model the time between independent events that happen at a constant average rate λ. Some examples of this distribution are:

  • The distance between one car passing by after the previous one.
  • The rate at which radioactive particles decay.

For the stochastic variable X, probability distribution function of it is:

 

and the cumulative distribution function is:

 

Exponential distribution is denoted as  , where m is the average number of events within a given time period. So if m=3 per minute, i.e. there are three events per minute, then λ=1/3, i.e. one event is expected on average to take place every 20 seconds.

MeanEdit

We derive the mean as follows.

 
 
 

We will use integration by parts with u=−x and v=e−λx. We see that du=-1 and dv=−λe−λx.

 
 
 
 

VarianceEdit

We use the following formula for the variance.

 
 
Failed to parse (syntax error): {\displaystyle \operatorname{Var}(X) = \int^\infin_{0}x^2 \2 e^{-\2 x} dx-{2}}

We'll use integration by parts with u=−x2 and v=e−2x. From this we have du=−2x and dv=−2e-2x

 
 

We see that the integral is just E[X] which we solved for above.

 
 

External linksEdit