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Beta Distribution — Definition, Formula & Examples

The beta distribution is a continuous probability distribution defined on the interval [0,1][0, 1], shaped by two positive parameters α\alpha and β\beta. It is commonly used to model proportions, probabilities, or any random variable naturally bounded between 0 and 1.

A random variable XX follows a beta distribution, written X∼Beta(α,β)X \sim \text{Beta}(\alpha, \beta), if its probability density function is f(x;α,β)=xα−1(1−x)β−1B(α,β)f(x; \alpha, \beta) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha, \beta)} for x∈[0,1]x \in [0,1] and α,β>0\alpha, \beta > 0, where B(α,β)=Γ(α)Γ(β)Γ(α+β)B(\alpha, \beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} is the beta function.

Key Formula

f(x;α,β)=xα−1(1−x)β−1B(α,β),E[X]=αα+β,Var(X)=αβ(α+β)2(α+β+1)f(x;\alpha,\beta) = \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha,\beta)}, \quad E[X] = \frac{\alpha}{\alpha+\beta}, \quad \text{Var}(X) = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}
Where:
  • xx = Value of the random variable, restricted to $[0, 1]$
  • α\alpha = First shape parameter (positive real number)
  • β\beta = Second shape parameter (positive real number)
  • B(α,β)B(\alpha,\beta) = Beta function, which normalizes the density so it integrates to 1

How It Works

The parameters α\alpha and β\beta control the shape of the distribution. When α=β\alpha = \beta, the distribution is symmetric around 0.5. When α>β\alpha > \beta, the distribution skews left (concentrating mass near 1), and when α<β\alpha < \beta, it skews right (concentrating mass near 0). The special case α=β=1\alpha = \beta = 1 gives a uniform distribution on [0,1][0,1]. In Bayesian statistics, the beta distribution serves as a conjugate prior for the Bernoulli and binomial likelihood, meaning that if your prior is beta and you observe binary data, the posterior is also beta with updated parameters.

Worked Example

Problem: Suppose a random variable X∼Beta(3,7)X \sim \text{Beta}(3, 7). Find the mean and variance of XX.
Step 1: Compute the mean using the formula E[X]=αα+βE[X] = \frac{\alpha}{\alpha + \beta}.
E[X]=33+7=310=0.3E[X] = \frac{3}{3+7} = \frac{3}{10} = 0.3
Step 2: Compute the variance using Var(X)=αβ(α+β)2(α+β+1)\text{Var}(X) = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}.
Var(X)=3⋅7(10)2⋅11=211100≈0.0191\text{Var}(X) = \frac{3 \cdot 7}{(10)^2 \cdot 11} = \frac{21}{1100} \approx 0.0191
Answer: The mean is 0.30.3 and the variance is approximately 0.01910.0191. Since α<β\alpha < \beta, the distribution is right-skewed, with most of its mass concentrated below 0.5.

Visualization

Why It Matters

The beta distribution is central to Bayesian inference, where it models uncertainty about unknown probabilities — for example, estimating a click-through rate or the probability a medical treatment works. It also appears in A/B testing, reliability engineering, and project scheduling (via the PERT method).

Common Mistakes

Mistake: Confusing the shape parameters: assuming larger α\alpha shifts mass toward 0.
Correction: Larger α\alpha relative to β\beta shifts mass toward 1, not 0. The mean αα+β\frac{\alpha}{\alpha+\beta} increases as α\alpha grows.