1
$\begingroup$

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?

$\endgroup$
  • 3
    $\begingroup$ 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. $\endgroup$ – StubbornAtom May 29 at 8:05
  • 2
    $\begingroup$ Relevant: stats.stackexchange.com/questions/4659/…. $\endgroup$ – StubbornAtom May 29 at 8:11
1
$\begingroup$

Conceptually, the two distributions are different views of the same model. The binomial distribution is the PMF of k successes given n independent events each with a probability p of success. The beta distribution is the PDF for p given n independent events with k successes.

Mathematically, when $\alpha=k+1$ and $\beta=n - k + 1$, the beta distribution and the binomial distribution are related by a constant factor: $$ Beta(p; \alpha, \beta) = (n+1) Binom(k; n, p) $$

This resource proves the relationship to the binomial distribution:

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 $\beta+n-y$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.