Incorrect formula for triangle density

[ksvanhorn]

Summary:

Section 21.1 ("Custom Probability Function" > "Examples" > "Triangle distribution") give an incorrect formula for the pdf of the triangle distribution.

Description:

The text gives for formula as

$$ \textsf{triangle}(y \mid \alpha,\beta) = \frac{2}{\beta - \alpha}\ \left(1 - \left|y - \frac{\alpha + \beta}{\beta - \alpha}\right|\right) $$

but this is wrong, as you can easily see by setting $\alpha = 2$ and $\beta = 8$. (The mode should be at $y=5$, but $(2+8)/(8-2) = 5/3$.)

Additional Information:

The correct formula is

$$ \textsf{triangle}(y \mid \alpha,\beta) = \frac{2}{\beta - \alpha}\ \left(1 - \frac{2}{\beta - \alpha}\left|y - \frac{\alpha + \beta}{2}\right|\right) $$

for $y\in [\alpha, \beta]$.

Current Version:

v2.32