Understanding the Beta Distribution: PDF, CDF, and Real‑World Applications

Definition

In probability theory, the Beta distribution (also called the B distribution) is a continuous probability distribution defined on the interval [0,1] with two shape parameters α and β.

Probability Density Function

The probability density function (PDF) of the Beta distribution is:

f(x;α,β)=\frac{x^{α-1}(1-x)^{β-1}}{B(α,β)} where B(α,β) is the Beta function.

Cumulative Distribution Function

The cumulative distribution function (CDF) of the Beta distribution is:

F(x;α,β)=I_{x}(α,β) where Iₓ is the regularized incomplete beta function.

Properties

For parameters α>1 and β>1, the mode of the Beta distribution is (α‑1)/(α+β‑2).

The mean and variance are respectively α/(α+β) and αβ/[(α+β)²(α+β+1)].

The skewness is (2(β‑α)√(α+β+1))/((α+β+2)√(αβ)).

The kurtosis (excess) is 6[(α‑β)²(α+β+1)‑αβ(α+β+2)]/[αβ(α+β+2)(α+β+3)].

Example

Relative humidity, the ratio of water vapor present in air to the maximum possible at a given temperature, ranges between 0 and 1 and exhibits randomness, making the Beta distribution a suitable model.

Studies such as “Extreme Climate in the Tarim Basin and Its Application in Oil‑field Engineering Design” have shown that both the daily maximum relative humidity in winter and the daily minimum relative humidity in summer in the Tarim Basin follow a Beta distribution.

References:

Baidu Baike entry on Beta distribution

Degroot, M.H., Li, Y., Li, J. “Beta distribution, multinomial distribution and bivariate normal distribution” Journal of Shaoyang College, 1994, 02:189‑193.

Xu, Chuan‑sheng. “Discussion on properties and applications of the Beta distribution” Journal of Linyi Normal University, 2001, 04:6‑8.