A continuous function vanishes on $(-\infty,a]$ and on $[c,\infty),$ its graph is a straight line on the interval $[a,b]$ and another straight line on $[b,c],$ and its integral is $1.$

The mean of the probability distribution whose density this is, is
$$
\frac{a+b+c} 3
$$
and the variance is
$$
\frac{a^2+b^2+c^2-ab-ac-bc} {18}.
$$
Both of these are symmetric functions of $a,b,c,$ despite the fact that the role of $b$ in the first paragraph above is different from those of $a$ and $c.$

Is this symmetry somehow surreptitiously present in the characterization of this distribution in the first paragraph above?