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Section 8.4 Gamma Distribution

Extending the exponential distribution model developed above, consider a Poisson Process where you start with an interval of variable length X so that X measures the interval needed in order to obtain the rth success for some natural number r. Then \(R = (0,\infty)\) and the resulting distribution of X will be called a Gamma distribution.

Definition 8.4.1 Gamma Function
\begin{equation*} \Gamma(t) = \int_0^{\infty} u^{t-1} e^{-u} du \end{equation*}

Letting n be a natural number and applying integration by parts one time gives

\begin{align*} \Gamma(n+1) & = \int_0^{\infty} u^n e^{-u} du\\ & = -u^n \cdot e^{-u} \big |_0^{\infty} + n \int_0^{\infty} u^{n-1} e^{-u} du \\ & = 0 - 0 + n \Gamma(n) \end{align*}

Continuing using an inductive argument to obtain the final result.

To find the probability function for the gamma distribution, once again focus on the development of F(x). Assuming r is a natural number greater than 1 and noting that X measures the interval length needed in order to achieve the rth success

\begin{align*} F(x) & = P(X \le x)\\ & = 1 - P(X \gt x)\\ & = 1 - P(\text{fewer than r successes in [0,x]})\\ & = 1 - \big [ \frac{(\lambda x)^0 e^{-\lambda x}}{0!} + \frac{(\lambda x)^1 e^{-\lambda x}}{1!} + ... + \frac{(\lambda x)^{r-1} e^{-\lambda x}}{(r-1)!} \big ]\\ & = 1 - \sum_{k=0}^{r-1} \frac{(\lambda x)^k e^{-\lambda x}}{k!} \end{align*}

where the discrete Poisson probability function is used on the interval [0,x]. The derivative of this function however is "telescoping" and terms cancel. Indeed,

\begin{align*} F'(x) & = \lambda e^{-\lambda x}/0!\\ & - \lambda e^{-\lambda x}/1! + \lambda x \cdot \lambda e^{-\lambda x}/1!\\ & - \lambda^2 2x e^{-\lambda x}/2! + \lambda^2 x^2 \cdot \lambda e^{-\lambda x}/2!\\ & - \lambda^3 3x^2 e^{-\lambda x}/3! + \lambda^3 x^3 \cdot \lambda e^{-\lambda x}/3!\\ & . . .\\ & - \lambda^{r-1} (r-1)x^{r-2} e^{-\lambda x}/(r-1)! + \lambda^{r-1} x^{r-1} \cdot \lambda e^{-\lambda x}/(r-1)!\\ & = \lambda^r x^{r-1} e^{-\lambda x}/(r-1)! \end{align*}

where you can replace \((r-1)! = \Gamma(r)\text{.}\)

Notice that for this random variable, \(\mu = \lambda T\) can be obtained for the exponential distribution. For the Gamma distribution, the following takes \(\mu\) to be the average interval till the first success and then modifies the corresponding Gamma parameters according to increasing values of r.

Definition 8.4.3 Gamma Distribution Probability Function
If X measures the interval until the rth success and \(\mu as the average interval until the 1st success\text{,}\) then X with probability function
\begin{equation*} f(x) = \frac{x^{r-1} \cdot e^{-x / \mu}}{\Gamma(r) \cdot \mu^r} \end{equation*}
has a Gamma Distribution.

Evaluate the sage code below.

Derivation of mean, variance, skewness, and kurtosis. Pick "alpha" for the general formulas.

Finally, the interactive cell below can be used to compute the distribution function for the gamma distribution for various input values. If you desire to let r get bigger than the slider allows, feel free to edit the cell above and evaluate again.