Gamma Distribution

Gamma Distribution#

The gamma distribution is given by,

\[ f_X(x; \lambda, r) = \frac{\lambda^r}{\Gamma(r)}x^{r-1}e^{-\lambda x}, \quad \text{for } x \ge 0 \]

Where \(\Gamma(r)\) is the gamma function given by,

\[ \Gamma(r) = \int_0^\infty x^{r-1}e^{-x} ~dx \]

Expectation : The expected value is given by, $\( E(X) = \frac{r}{\lambda} \)$

Variance : The variance is given by,

$$
\text{SD}(X) = \frac{\sqrt{r}}{\lambda}
$$

Sums : The sum of iid gamma distributions have the distribution

* For fixed rate $\lambda$

    $$
    \sum_k^n X_k \sim \text{gamma}\left(\sum_k^n r_k, \lambda \right)
    $$

MGF : $\( M_X(t) = \left( \frac{\lambda}{\lambda-t} \right)^r, \quad \text{for } t < \lambda \)$

Rate Parameter#

The rate parameter \(\lambda\) is the scale parameter of the distribution

Shape Parameter#

The parameter \(r\) is known as the shape parameter of the distribution.

  • \(r=1\) gives the exponential distribution

  • \(r \to \infty\) gives the normal distribution

Gamma Function#

The gamma function,

\[ \Gamma(r) = \int_0^\infty x^{r-1}e^{-x} ~dx \]

has solutions that follow,

\[ \Gamma(r+1) = r\Gamma(r) \]

Which implies two sets of solutions:

  • For positive integer \(r\), $\( \Gamma(r) = (r-1)! \)$

  • For non-integers,

    • For rationals of \(\frac{1}{2} + r\) for \(r \ge 0\) $\( \Gamma\left(\frac{1}{2} + r\right) = \begin{cases} \dfrac{1}{\sqrt{\pi}}\\ \prod_\limits{k=1}^{r} \left(\frac{1}{2} + k - 1\right)\dfrac{1}{\sqrt{\pi}} \end{cases} \)$

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