Gamma Distribution

This is the opposite of Poisson Distribution. In poisson, we measure how many rare events would happen in a fixed continuous interval.

In Gamma, we measure how much continuous interval would require for a fixed number of rare events to occur.

Since the random variable is the Continuous Interval, this is a Continuous Distribution.

Gamma distribution is defined using two different parameterization methods.

1. Shape-Rate parameterization ($\alpha, \lambda$)
2. Shape-Scale parameterization ($\alpha,\beta$) $\beta=\frac{1}{\lambda}$

This course will forward with the shape-scale parameterization

PDF

$$f(x; \alpha, \beta) = \frac{1}{\beta^\alpha\Gamma(\alpha)} x^{\alpha-1} e^{-\frac{x}{\beta}}, \quad x > 0$$

Interactive Graph - https://www.desmos.com/calculator/vk2tqrxpk5

Parameters & Key Properties:

• Shape ($\alpha > 0$): Controls the profile and skewness.
• Scale ($\beta > 0$): Controls the horizontal scale/stretch along the $x$-axis.
• Mean: $\mathbb{E}[X] = \alpha\beta$
• Variance: $\text{Var}(X) = \alpha\beta^2$
• Scaling Factor for Area to be 1 : $\frac{1}{\Gamma(\alpha)\beta^\alpha}$

Derivation of $k$ (Constant Factor)

We need the $k$ factor to keep the area always $1$ for given $\alpha,\beta$ values Starting with generic Gamma based PDF,

\begin{align*} \int_{0}^{\infty} k x^{\alpha-1} e^{-\frac{x}{\beta}} \, dx &= 1 \\ k \int_{0}^{\infty} x^{\alpha-1} e^{-\frac{x}{\beta}} \, dx &= 1 \\ u = \frac{x}{\beta} \implies x = \beta u \implies dx = \beta \, du \\ \int_{0}^{\infty} (\beta u)^{\alpha-1} e^{-u} (\beta \, du) &= \int_{0}^{\infty} \beta^{\alpha-1} u^{\alpha-1} e^{-u} \beta \, du \\ &= \beta^{\alpha} \int_{0}^{\infty} u^{\alpha-1} e^{-u} \, du \\ &= \beta^{\alpha} \Gamma(\alpha) \\ \\ \end{align*}$$

\begin{align*} k \cdot \beta^{\alpha} \Gamma(\alpha) &= 1 \ k &= \frac{1}{\Gamma(\alpha) \beta^{\alpha}} \end{align*}

## Shape Vs Scale 1. $\alpha$ - Changes shape/profile low $\alpha$ - skewed to left high $\alpha$ - more even like normal distribution Shape determines **how many** events we waiting for 2. $\beta$ - Changes horizontal stretch low $\beta$ - less stretch high $\beta$ - more stretch Scale determines **how many intervals** between 2 events | $\alpha$ change | $\beta$ change | | ------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------------------------ | | ![[gamma-alpha.png]] | ![[gamma-beta.png]] | | Different profiling<br>$\alpha$= 1 - no left<br>2 - huge left skew<br>3 - less huge left skew<br>5 - more into even<br>10 - more even | Same profiling different stretch<br>$\beta$= 1 - tiny stretch<br>2 - a bit stretched<br>5 - more stretched<br>10 - further stretched | ## Properties - Gamma distribution has a additive property similar to [[Poisson Distribution]] Only when the **Scale** parameter is same.

\text{If } X_1 \sim \text{Gamma}(\alpha_1, \beta) \quad \text{and} \quad X_2 \sim \text{Gamma}(\alpha_2, \beta)

$$$$

\implies X_1 + X_2 \sim \text{Gamma}(\alpha_1 + \alpha_2, , \beta)

$$$$