Ex:           An engineer is measuring the illumination, Z, at one point in an optical circuit. The engineer knows that Z is a linear combination of illumination values, X and Y, at two other points in the circuit. In other words, Z = aX + bY. The engineer has gathered the following information about the illumination values:

                                              

                                                               

Find the values of a and b, (assuming they are positive).

Sol'n:      We use the following tools for linear combinations of random variables:

Substituting values given in the problem, we have the following two equations with unknowns a and b:

We can solve the first equation for b and substitute the result into the second equation:

When we substitute for b, the second equation becomes a quadratic equation in a:

Multiplying both sides by 302 and combining terms gives the following quadratic equation:

The solutions are a = 1/4 or a = −3/38. We use the positive root, a = 1/4, as instructed in the problem.

Returning to the equation relating b to a, we get the following value: