Beta Distribution

Introduction to the beta distribution

1.The beta distribution is a continuous probability distribution on the open interval $(0, 1).$ It has two positive parameters $\alpha$ and $\beta,$ which determine the curvature of the density and the distribution of its mass along the interval. A beta-distributed random variable takes values only between $0$ and $1,$ so the distribution can model proportions, ratios, and probabilities. Its probability density function is:

$$f(x; \alpha, \beta) = \frac{x^{\alpha - 1}(1 - x)^{\beta - 1}}{B(\alpha, \beta)} \quad 0 < x < 1$$

2.The normalizing constant $B(\alpha, \beta)$ is the beta function, which has the gamma-function representation:

$$B(\alpha, \beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha + \beta)}$$

3.Substitution gives the following equivalent form of the density:

$$f(x; \alpha, \beta) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha - 1}(1 - x)^{\beta - 1}$$

4.Outside the interval $(0, 1),$ the density is zero. The gamma function $\Gamma(c),$ defined for every $c \in \mathbb{R}^+,$ has the improper integral representation:

$$\Gamma(c) = \int_{0}^{+\infty} x^{c - 1} e^{-x} \ dx$$

5.For every natural number $n,$ the identity $\Gamma(n + 1) = n!$ relates the gamma function to the factorial. The function $x \mapsto \Gamma(x + 1)$ therefore extends the factorial to positive real numbers. The cumulative distribution function is $F(x) = P(X \le x).$ For $0 \le x \le 1,$ it has the integral representation:

$$F(x; \alpha, \beta) = \frac{1}{B(\alpha, \beta)} \int_{0}^{x} t^{\alpha - 1}(1 - t)^{\beta - 1} \ dt = I_x(\alpha, \beta)$$

6.The function $I_x(\alpha, \beta)$ is the regularized incomplete beta function, the incomplete integral divided by the complete beta function $B(\alpha, \beta).$ For positive integer parameters it equals an upper tail of a binomial distribution with $\alpha + \beta - 1$ trials:

$$I_x(\alpha, \beta) = \sum_{j = \alpha}^{\alpha + \beta - 1} \binom{\alpha + \beta - 1}{j} x^{j}(1 - x)^{\alpha + \beta - 1 - j}$$

The shape of the beta distribution

7.The beta distribution has different shapes according to the values of $\alpha$ and $\beta.$ Its density can be unimodal, U-shaped, or monotonic. When both parameters are greater than $1$ or both are less than $1,$ the density has the interior critical point:

$$x_0 = \frac{\alpha - 1}{\alpha + \beta - 2}$$

8.The nature of this point depends on the parameters.

10.Exchanging the two parameters reflects the density about the midpoint of the interval. If $X \sim \mathrm{Beta}(\alpha, \beta),$ then $1 - X \sim \mathrm{Beta}(\beta, \alpha).$ When $\alpha = \beta,$ this reflection leaves the density unchanged, so it is symmetric about the vertical line $x = \tfrac{1}{2}.$

11.The figure shows the symmetric case $0 < \alpha = \beta < 1.$ The density is U-shaped and tends to infinity near the endpoints of the interval $(0, 1).$ It has a minimum at $x = x_0,$ which is its lowest value on the domain.

12.When $\alpha = \beta > 1,$ the beta distribution is symmetric about the vertical line $x = \tfrac{1}{2}$ and has its single mode there. As the common parameter increases, the density becomes more concentrated around $x = \tfrac{1}{2},$ and the normal distribution gives a closer approximation.

13.For large values of $\alpha$ and $\beta,$ the approximating normal distribution has the following mean and variance:

$$\mu = \frac{\alpha}{\alpha + \beta} \qquad \sigma^2 = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}$$

14.In the symmetric case $\alpha = \beta = k,$ these parameters are:

$$\mu = \tfrac{1}{2} \qquad \sigma^2 = \frac{1}{4(2k + 1)} \approx \frac{1}{8k}$$

15.Thus, as $k \to \infty,$ the distributional approximation is:

$$\mathrm{Beta}(k, k) \approx \mathcal{N}\!\left(\tfrac{1}{2}, \frac{1}{4(2k + 1)}\right)$$

16.For $X \sim \mathrm{Beta}(\alpha, \beta),$ the probability density function, mean, variance, and standard deviation are:

Mean of the beta distribution

18.A beta-distributed random variable has a mean, or expected value, determined by the shape parameters $\alpha$ and $\beta.$ By definition, its mean is:

$$\mu = E(X) = \int_{0}^{1} x f(x; \alpha, \beta) \ dx$$

19.Substituting the probability density function gives:

$$E(X) = \int_{0}^{1} x \frac{x^{\alpha - 1} (1 - x)^{\beta - 1}}{B(\alpha, \beta)} \ dx$$

20.Combining the powers of $x$ gives:

$$E(X) = \frac{1}{B(\alpha, \beta)} \int_{0}^{1} x^{\alpha} (1 - x)^{\beta - 1} \ dx$$

21.The integral on the right-hand side is the beta function $B(\alpha + 1, \beta),$ hence:

$$E(X) = \frac{B(\alpha + 1, \beta)}{B(\alpha, \beta)}$$

22.The beta function has the gamma-function representation:

$$B(\alpha, \beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha + \beta)}$$

23.Since the gamma function has the recurrence $\Gamma(c + 1) = c\Gamma(c),$ the ratio simplifies to:

$$E(X) = \frac{\alpha}{\alpha + \beta}$$

24.

The mean depends only on the two shape parameters. It exceeds $\tfrac{1}{2}$ when $\alpha > \beta$ and falls below $\tfrac{1}{2}$ when $\alpha < \beta.$

25.For each integer $k \geq 1,$ the same integral with $x^k$ in place of $x$ gives the corresponding moment of the distribution:

$$E(X^k) = \frac{B(\alpha + k, \beta)}{B(\alpha, \beta)} = \prod_{r = 0}^{k - 1} \frac{\alpha + r}{\alpha + \beta + r}$$

26.The mean is the case $k = 1,$ and the second moment $E(X^2),$ used in the next section for the variance, is the case $k = 2.$

Variance of the beta distribution

27.The variance of the beta distribution is the expected squared deviation from the mean. Equivalently, it has the second-moment formula:

$$\sigma^2 = \mathrm{Var}(X) = E(X^2) - [E(X)]^2$$

28.The second moment has the integral representation:

$$ \begin{align} E(X^2) &= \int_{0}^{1} x^2 f(x; \alpha, \beta) \ dx \\[6pt] &= \frac{1}{B(\alpha, \beta)} \int_{0}^{1} x^{\alpha + 1} (1 - x)^{\beta - 1} \ dx \\[16pt] &= \frac{B(\alpha + 2, \beta)}{B(\alpha, \beta)} \end{align} $$

29.Substituting this expression and the mean into the definition gives:

$$\sigma^2 = \frac{B(\alpha + 2, \beta)}{B(\alpha, \beta)} - \left(\frac{\alpha}{\alpha + \beta}\right)^2$$

30.The beta-gamma identity is:

$$B(\alpha, \beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha + \beta)}$$

31.Together with the recurrence $\Gamma(c + 1) = c\Gamma(c),$ this identity simplifies the variance to:

$$\sigma^2 = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}$$

32.

If the ratio $\alpha/(\alpha + \beta)$ is fixed, increasing the concentration $\alpha + \beta$ decreases the variance and concentrates the distribution around its mean.

Special cases and related distributions

33.Particular values of the shape parameters reduce the beta distribution to other named distributions on the interval $(0, 1).$

34.The uniform distribution is the case $\alpha = \beta = 1.$ Its probability density function is constant on the interval $(0, 1).$ More generally, the continuous uniform distribution on an interval $(a, b)$ has the density:

$$ f(x) = \begin{cases} \dfrac{1}{b - a} & a < x < b \\[6pt] 0 & \mathrm{otherwise} \end{cases} $$

35.For $a = 0$ and $b = 1,$ this reduces to $f(x) = 1,$ which is exactly the $\mathrm{Beta}(1, 1)$ density, constant across $(0, 1).$

36.For $\alpha = \beta = \tfrac{1}{2},$ the beta distribution is the arcsine distribution. Its density is:

$$f\!\left(x; \tfrac{1}{2}, \tfrac{1}{2}\right) = \frac{1}{\pi \sqrt{x(1 - x)}}$$

37.This density diverges at both endpoints and has its minimum at $x = \tfrac{1}{2},$ as in the U-shaped case described above. The values $\alpha = 2$ and $\beta = 1$ give a triangular density that increases linearly on the interval:

$$f(x; 2, 1) = 2x$$

38.After exchanging the parameters, $\alpha = 1$ and $\beta = 2$ give the reflected density $f(x; 1, 2) = 2(1 - x).$ The identity $1 - X \sim \mathrm{Beta}(\beta, \alpha)$ maps one case to the other.

39.The beta distribution also arises from the gamma distribution. If $U$ and $V$ are independent gamma variables with shape parameters $\alpha$ and $\beta$ and the same scale, their normalized ratio is beta-distributed:

$$\frac{U}{U + V} \sim \mathrm{Beta}(\alpha, \beta)$$

40.The change of variables from $(U, V)$ to $(U/(U + V), U + V)$ gives this distributional identity and the relation between the beta and gamma functions.