Let, x1 <  = x2 <  = x3....... <  = xn

and

p1 + p2 + p3 + ....... + pn = 1

We all know that average of x1, x2, x3......., xn is in [x1,xn] and it is easy to understand.

In a contest, I assumed Expected value = p1 * x1 + p2 * x2 + p3 * x3 + ....... + pn * xn is in [x1,xn] regardless how probability is distributed that means the sum of probability can be 1 in many different ways.

My assumption was right and got ac. I'm interested to know the proof.

TIA