Statistics/Distributions/Exponential

Exponential Distribution Exponential continuous 325px|Probability density function 325px|Cumulative distribution function λ > 0 rate, or inverse scale x ∈ [0, ∞) λ e−λx 1 − e−λx λ−1 λ−1 ln 2 0 λ−2 2 6 1 − ln(λ) 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. Mean 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. Variance We use the following formula for the variance. We'll use integration by parts with and . From this we have and . We see that the integral is just which we solved for above. External links Interactive Exponential Distribution Web Applet (Java)