1. The Exponential and Gamma Distributions

2. Non-symmetric distributions
The density curve of the normal distribution is symmetric and bell-shaped. There are many practical situations where the variable of interest is skewed (not symmetric).
In this section we consider a family of distributions (gamma) with this property, beginning with a special case (the exponential distribution).

3. Definition of the exponential distribution
X is said to have an exponential distribution with parameter if the pdf of X is

4. Mean and variance
The expected value can be computed using integration by parts with
and , the result being .
The variance can be computed by integrating by parts twice, with the result

5. Cumulative distribution function
The exponential cdf is

6. Poisson process and the exponential distribution
The exponential distribution is related to the Poisson process.
For example, if is the time until the first event,

7. Application of the exponential distribution
If the number of events occurring in any interval of length t has Poisson distribution with parameter , and the number of occurrences in non-overlapping intervals is independent, then the distribution of the elapsed time between the occurrence of two successive events is exponential with parameter

8. Example
Suppose that calls are received at a 24-hour “suicide hotline” according to a Poisson process with rate calls per day.
Then the number of days between successive calls has an exponential distribution with parameter .5, so that the probability that more than two days elapse between calls is
The expected time between calls is 1/.5=2 days.

9. Memoryless property For the exponential distribution,
This is the same as Thus the exponential distribution is used to model the lifetime of components that show no effect of age.

10. The Gamma function
Before defining the family of gamma distributions, we introduce the gamma function.
For any , the gamma function is defined by

11. Important properties of the gamma function
From the definition of the integral,
For any ,
For any positive integer n,

12. The Gamma distribution
A continuous random variable X is said to have a gamma distribution if the pdf of X is
where the parameters satisfy

13. Properties of the gamma distribution
The standard gamma distribution has is called a scale parameter (values other than
stretch or compress the distribution in the x direction.)
The exponential distribution results from taking and

14. Mean and variance of gamma distribution
The mean of the distribution is
The variance of the distribution is

15. The chi-square distribution
Let be a positive integer. Then the rv X has a chi-square distribution if the pdf of X is the gamma distribution with and The pdf is
is the number of degrees of freedom (df) of X.