Question

Question:In Example 3.4.5, compute the probability that water demandXis greater than electric demandY.

Short answer

The probability that water demandXis greater than electric demandY is 0.6350.

Step-by-step solution

Given information

Referring to example 3.4.5, the joint probability of water demand X and the electric demand Y is as follows,

\(f\left( {x,y} \right) = \left\{ \begin{array}{l}\frac{1}{{29204}}\;if\;4 \le x \le 200\;and\;1 \le y \le 150\\0\;otherwise\end{array} \right.\)

Define the region

Referring to example 3.4.5, considering Y<X and Y<150, the region of X and Y can be written as \(\left\{ {\left( {x,y} \right):4 < x < 200,1 < y < \min \left( {x,150} \right)} \right\}\).

Calculate the probability

The probability that the water demand is greater than electric demand is,

\(\begin{array}{c}\Pr \left( {X > Y} \right) = \int\limits_4^{200} {\int\limits_1^{\min \left( {x,150} \right)} {f\left( {x,y} \right)dydx} } \\ = \int\limits_4^{200} {\int\limits_1^{\min \left( {x,150} \right)} {\frac{1}{{29204}}dydx} } \\ = \int\limits_4^{200} {\frac{{\min \left\{ {\left( {x - 1} \right),149} \right\}}}{{29204}}dx} \\ = \int\limits_4^{150} {\frac{{\left( {x - 1} \right)}}{{29204}}dx + \int\limits_{150}^{200} {\frac{{149}}{{29204}}dx} } \\ = \left. {\frac{{{{\left( {x - 1} \right)}^2}}}{{2 \times 29204}}} \right|_4^{150} + \frac{{50 \times 149}}{{29204}}\\ = \frac{{{{149}^2} - {3^2}}}{{58408}} + \frac{{7450}}{{29204}}\\ = 0.6350\end{array}\)

Thus, the required probability is 0.6350