Solved – Mean and variance of a Beta distribution with $\alpha \ge 1$ or $\beta \ge 1$

beta distribution

What conditions must satisfy the mean and variance of a Beta distribution so that the parameters $\alpha,\beta$ are not both less than 1?

Best Answer

The parameters of a $\text{Beta}(\alpha,\beta)$ distribution with mean $0\lt m\lt 1$ and variance $0\lt v\lt m(1-m)$ are

$$\alpha = m\frac{m(1-m)- v}{v},\quad \beta = (1-m)\frac{m(1-m)-v}{v}.$$

This shaded contour plot of $\alpha$ has contours ranging from $0$ (at the top of the colored region) to $1$ (along the bottom). The plot of $\beta$ is its mirror image.

If they are not both less than $1$, then algebra shows us that

$$v \le m\max\left(\frac{m(1-m)}{1+m}, \frac{(1-m)^2}{2-m}\right).$$

The valid set of all possible means and variances of Beta distributions is contained beneath the gray curve. Within that set, those where one or both of $\alpha$ and $\beta$ are $1$ or greater is shown in darker blue. These tend to have lower variances on the whole.

Related Solutions

Beta Distribution – Calculating Parameters using Mean and Variance

I set$$\mu=\frac{\alpha}{\alpha+\beta}$$and$$\sigma^2=\frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)}$$and solved for $\alpha$ and $\beta$. My results show that$$\alpha=\left(\frac{1-\mu}{\sigma^2}-\frac{1}{\mu}\right)\mu^2$$and$$\beta=\alpha\left(\frac{1}{\mu}-1\right)$$

I've written up some R code to estimate the parameters of the Beta distribution from a given mean, mu, and variance, var:

estBetaParams <- function(mu, var) {
  alpha <- ((1 - mu) / var - 1 / mu) * mu ^ 2
  beta <- alpha * (1 / mu - 1)
  return(params = list(alpha = alpha, beta = beta))
}

There's been some confusion around the bounds of $\mu$ and $\sigma^2$ for any given Beta distribution, so let's make that clear here.

$\mu=\frac{\alpha}{\alpha+\beta}\in\left(0, 1\right)$
$\sigma^2=\frac{\alpha\beta}{\left(\alpha+\beta\right)^2\left(\alpha+\beta+1\right)}=\frac{\mu\left(1-\mu\right)}{\alpha+\beta+1}<\frac{\mu\left(1-\mu\right)}{1}=\mu\left(1-\mu\right)\in\left(0,0.5^2\right)$

Solved – Beta distribution vs beta binomial distribution: alpha and beta

Beta and Beta-binomial are two different distributions. Beta is a continuous distribution (values in the $[0, 1]$ range) while Beta-binomial is discrete (integers from $0$ to $n$). So it is not a good idea to use both distributions to approximate your data since it can be discrete or continuous, but not both.

Best Answer

Related Solutions

Beta Distribution – Calculating Parameters using Mean and Variance

Solved – Beta distribution vs beta binomial distribution: alpha and beta

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