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

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?

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 $$\frac{4}{3}$$.