# Is a Beta distribution a continuous version of the Binomial Theorem?

The visual appearance of the PDF for a Beta distribution resembles that for the terms in the Binomial Theorem. Is the former a continuous variant of the discrete terms of the latter? Are they related?

Here's what I have so far:

\begin{align} \textrm{PDF Beta Distribution} &= \frac{x^{\alpha-1}(1-x)^{\beta-1}}{B(\alpha, \beta)} \\ &= \frac{\Gamma(\alpha+\beta)x^{\alpha-1}(1-x)^{\beta-1}}{\Gamma(\alpha)\Gamma(\beta)} \\ &= \frac{(\alpha+\beta-1)!x^{\alpha-1}(1-x)^{\beta-1}}{(\alpha-1)!(\beta-1)!} \\ &= \alpha \binom{\alpha+\beta-1}{\alpha} x^{\alpha-1}(1-x)^{\beta-1} \end{align} compared with the k-th term of the expansion of $$(x + y)^n$$:

$$\binom{n}{k}x^{n-k}y^k$$ now setting $$y=(1-x)$$, $$k=\alpha$$, and $$n=\alpha+\beta-1$$ gives:

$$\binom{\alpha+\beta-1}{\alpha}x^{\beta-1}(1-x)^\alpha$$ The dots aren't quite connected but do suggest a connection.

So are the terms of a binomial expansion related to the discrete steps for a Beta- distribution where $$\alpha$$ and $$\beta$$ are natural numbers?

• Beta and Binomial are intrinsically related; notably their distribution functions can be expressed in terms of each other. And as mentioned in this wiki page, Beta is a conjugate distribution of Binomial. – StubbornAtom May 29 at 8:05
• Relevant: stats.stackexchange.com/questions/4659/…. – StubbornAtom May 29 at 8:11

Mathematically, when $$\alpha=k+1$$ and $$\beta=n - k + 1$$, the beta distribution and the binomial distribution are related by a constant factor: $$Beta(p; \alpha, \beta) = (n+1) Binom(k; n, p)$$
Suppose $$\mathit{X}$$ is a random variable having a Beta distribution with parameters $$\alpha$$ and $$\beta$$. Let $$\mathit{Y}$$ be another random variable such that its distribution conditional on $$\mathit{X}$$ is a binomial distribution with parameters $$n$$ and $$\mathit{X}$$. Then, the conditional distribution of $$\mathit{X}$$ given $$\mathit{Y}=y$$ is a Beta distribution with parameters $$\alpha+y$$ and $$\beta+n-y$$.