Gamma, Exponential And Beta Distributions

1. Core Concept Overview & Intuition

The gamma, exponential, and beta distributions form a family of continuous probability distributions defined on restricted supports and governed by positive shape and rate parameters. They are central to modelling waiting times, service durations, and proportions, and their moments are obtained by direct integration against the density or by differentiating the moment generating function.

The gamma distribution is a two-parameter family on the positive half-line. Its parameter $\alpha$ controls the shape of the density and its parameter $\beta$ controls the rate at which the density decays. Because the density is supported on $(0,\infty)$, the gamma law is a natural model for non-negative quantities such as lifetimes and accumulated waiting times.

The exponential distribution is the special case of the gamma law in which the shape parameter equals one. It is the simplest continuous waiting-time model and is characterized by a single positive rate parameter $\lambda$. Its density decreases monotonically from $\lambda$ at the origin, so small waiting times are more probable than large ones.

The beta distribution is a two-parameter family supported on the unit interval $(0,1)$. Because its support is bounded, it is the natural model for proportions and fractions. The two positive parameters $\alpha$ and $\beta$ jointly determine the shape of the density, and the special choice $\alpha=\beta=1$ produces the uniform distribution over $(0,1)$.

A unifying theme across all three families is that their moments are computed by the same programme: form the integral of $x^{k}$ against the density, recognize the integrand as the kernel of a gamma or beta integral, and read off the answer using the gamma function identities. The moment generating function provides an alternative route to the mean and variance by repeated differentiation at $t=0$.

2. Mathematical Formalization & Core Definitions

2.1 Gamma Distribution

> Definition (Gamma Distribution). A continuous random variable $X$ is said to have the gamma distribution with parameters $\alpha$ and $\beta$, where $\alpha>0$ and $\beta>0$, if its probability density function is of the form

$$
f(x)=\begin{cases}\dfrac{\beta^{\alpha}}{\Gamma(\alpha)}x^{\alpha-1}e^{-\beta x}, & x>0,
0, & \text{otherwise}.\end{cases}
$$

> We write $X\sim\Gamma(\alpha,\beta)$ to mean that $X$ has the gamma distribution with parameters $\alpha$ and $\beta$.

Here $\alpha$ is the shape parameter, $\beta$ is the rate parameter, and $\Gamma(\cdot)$ denotes the gamma function, which satisfies the recursion $\Gamma(\alpha+1)=\alpha\Gamma(\alpha)$ for $\alpha>0$.

2.2 Exponential Distribution

> Definition (Exponential Distribution). If a random variable $X$ has density given by

$$
f(x)=\begin{cases}\lambda e^{-\lambda x}, & x>0,\ \lambda>0,
0, & \text{otherwise},\end{cases}
$$

> then $X$ is defined to have an exponential distribution with rate parameter $\lambda$.

The exponential density is the gamma density with $\alpha=1$ and $\beta=\lambda$, since $\Gamma(1)=1$ and $x^{\alpha-1}=x^{0}=1$.

2.3 Beta Distribution

> Definition (Beta Distribution). A continuous random variable $X$ is said to have the beta distribution with parameters $\alpha$ and $\beta$, where $\alpha>0$ and $\beta>0$, if its probability density function is of the form

$$
f(x)=\begin{cases}\dfrac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}x^{\alpha-1}(1-x)^{\beta-1}, & 0<x<1,
0, & \text{otherwise}.\end{cases}
$$

The normalizing constant is expressed through the beta function, defined by the identity

$$
\int_{0}^{1}x^{\alpha-1}(1-x)^{\beta-1}\,dx=\frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)}=B(\alpha,\beta).
$$

This identity is the key computational device for all moments of the beta random variable.