Solved – Beta distribution from mean and quantile

Suppose I'm given the mean and one quantile (e.g. the 20% quantile) of a random variable $x$, and I want to find the parameters $alpha$ and $beta$ of a Beta distribution that has the same mean and quantile. Is there an efficient way to do it?

Using just the mean, I know that since $bar{x} = frac{alpha}{alpha+beta}$, we have $beta = frac{alpha}{bar{x}} – alpha$. So we only really have one parameter to estimate. But I'm unsure how to use the quantile information to take the next step. Maybe there's something I can do with the Incomplete Beta when I know the ratio $frac{beta}{alpha} = frac{1-bar{x}}{bar{x}}$?

I have access to R myself, so I could use a numerical optimizer for this, but ideally I need a method that can be carried out in Excel in someone else's environment. Excel does have BETA.DIST() and BETA.INV() functions available. A look-up table would be fine, but a closed-form formula would be better if it's possible.

If you really have to do it with pesky Excel:

  1. Create cells with quantile probability $p$, quantile value $q$, mean $m$.

  2. Create a cell with some initial $alpha$ value. Create a cell with formula $beta=left(frac{1-m}{m}right)alpha$.

  3. Create a cell with formula $mathrm{abs}(q – mathrm{beta.inv}(p, alpha,beta))$.

  4. Go to "Data" > "What-If Analysis" > "Goal Seek". Choose the previous cell for item "Set cell", put $0$ in "To value", and choose the $alpha$ cell for "By changing cell". Press "OK".

  5. Next time: Use R! (I'm joking. I know you're an R user.)

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