Question

Water flows in and out of a dam such that the daily inflow is uniform on [0,2](megaliters) and the daily outflow is uniform on [0,1], independent of theinflow. Each day the surplus water (if there is any) is collected for an irrigationproject. Compute the expected amount of surplus water in a given day

Answer 1

Given,

The daily inflow of water, X is,
$X \sim U[0,2]$

The daily outflow of water, Y is,
$Y \sim U[0,1]$

The daily surplus of water, S would be given as,

To Compute the expected amount of surplus water in a given day,

$E(D) = E(X - Y) = E(X) - E(Y) = \frac{0+2}{2} - \frac{0+1}{2} = 1 - 0.5 = \frac{1}{2}$

Answer : Required expected value of surplus water is 0.5 megalitres.