The Beta Distribution

The beta lives on $(0,1)$, so it models proportions: the fraction of a claim paid, a loss ratio, a probability that is itself uncertain. The exam hands you the density and asks you to integrate it, the one skill worth drilling.

$X \sim \text{Beta}(a, b)$ on the interval $(0, 1)$:

$$f(x) = \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}\,x^{a-1}(1-x)^{b-1}, \quad 0 < x < 1$$

The leading constant is whatever makes the density integrate to 1. The moments are simple ratios:

$$E[X] = \frac{a}{a+b}, \quad \text{Var}(X) = \frac{ab}{(a+b)^2(a+b+1)}$$

HIGH-FREQUENCY: The exam always gives you the density, often without naming it "beta." Recognize the $x^{a-1}(1-x)^{b-1}$ shape, read off $a$ and $b$, and either use the moment formulas or integrate directly. For integer powers the integral is elementary.

Since the support is $(0,1)$ and the density is a polynomial (for integer $a, b$), any moment is a routine polynomial integral. To get $E[X]$ you integrate $x\,f(x)$; to get a probability $P(X \le c)$ you integrate $f(x)$ from 0 to $c$.

Common mistakes

• Not recognizing the beta from its density. A density proportional to $x^{a-1}(1-x)^{b-1}$ on $(0,1)$ is beta. The exam often omits the name.
• Misreading the powers. $f(x) = 12x^2(1-x)$ has $a - 1 = 2$ (so $a = 3$) and $b - 1 = 1$ (so $b = 2$), not $a = 2$, $b = 1$.
• Using $E[X]^2$ where $E[X^2]$ is needed. The variance needs the second moment $E[X^2] = \int x^2 f(x)\,dx$, a separate integral from the mean.

Bottom line

• Beta($a, b$) on $(0,1)$: density proportional to $x^{a-1}(1-x)^{b-1}$.
• Mean: $E[X] = a/(a+b)$. Variance: $\text{Var}(X) = ab/[(a+b)^2(a+b+1)]$, or get it from $E[X^2] - E[X]^2$.
• Read $a, b$ off the density: the power on $x$ is $a-1$; the power on $1-x$ is $b-1$.
• A probability is $\int_0^c f(x)\,dx$, integrated only over $[0,c] \cap (0,1)$ since the density is 0 outside the unit interval.

Exam shortcut

Spot the $x^{a-1}(1-x)^{b-1}$ shape, read $a$ and $b$, then either apply $E = a/(a+b)$ or integrate directly. For probabilities, integrate the polynomial density over $[0, c]$, it is always elementary for integer parameters. "Powers plus one give $a$ and $b$." "Mean leans toward the bigger parameter." Symmetric only when $a = b$.

The full lesson (about 1,711 words, 11 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 2a
• 2b
• 2c
• 2d
• 2e
• 2f

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