Find average value of the function $f(x,y,z)=3x-4y+5z$ over the triangle (simplex) $x+y+z=1$ ($0\leq x,y,z<1$).

Question

Find the average value of the function $$f(x,y,z) = 3x-4y+5z$$ over the triangle (simplex) $\left\{ (x,y,z) \mid x+y+z=1 \land 0 \leq x,y,z < 1 \right\}$.

Is there a simple way to do this problem?

Answer 1

Let $f_1 = x, f_2 = y, f_3 = z$.

Show that $E[f_1] = E[f_2] = E[f_3]$ by symmetry.

Show that $ E[ f_1 + f_2 + f_3 ] = 1$ by linearity of expectation.

Hence, conclude that the value is $ \frac{4}{3}$.