Beta Distribution

#ProbabilityTheory

We have a random variable $X$ has a beta distribution with parameters $a > 0$ and $b > 0$ , denoted as $X \sim B (a, b)$ or $X \sim beta (a, b)$ , when its pdf is

f (x; a, b) = {\begin{cases} \frac{1}{B (a, b)} x^{a - 1} (1 - x)^{b - 1} & 0 \leq x \leq 1 \\ 0 \end{cases}

Where in this context, $B (a, b)$ represents the beta function.

We can calulate the cdf as

F (x) = {\begin{cases} 0 & x \leq 0 \\ \frac{1}{B (a, b)} \int_{0}^{x} u^{a - 1} (1 - u)^{b - 1} d u & 0 < x < 1 \\ 1 & x \geq 1 \end{cases}

where the integral above can be simplified into

F (x) = {\begin{cases} 0 & x \leq 0 \\ I_{x} (a, b) & 0 < x < 1 \\ 1 & x \geq 1 \end{cases}

where $I$ is the regularised incomplete beta function

We can get that

$E [X] = \frac{a}{a + b}$
$Var [X] = \frac{a b}{(a + b + 1) (a + b)^{2}}$
The mode is $\frac{a - 1}{a + b - 2}$
The median is $I_{1 / 2}^{[- 1]} (a, b)$ , where this is the inverse of the regularised incomplete beta function, this is dumb, but we can estimate it if $a, b \geq 1$ , then $$ \operatorname{median} \approx \frac{a-\frac{1}{3}}{a+b-\frac{2}{3}} $$
Let $X$ and $Y$ be independent random variables with distribution $Γ (a, λ)$ and $Γ (b, λ)$ , respectively. Then$$ \frac{X}{X+Y} \sim \text{B}(a,b) $$
We can calculate the $n$ th moment of $X$ as

E [X^{n}] = \frac{B (a + n, b)}{B (a, b)} = \frac{a^{\overset{―}{n}}}{(a + b)^{\overset{―}{n}}}

where $a^{\overset{―}{n}}$ represents the rising factorial of $a$