Ex:           The width of metal lines on an integrated circuit varies according to a distribution that falls off linearly above a certain minimum width, (which is x = 0). Find the expected value for excess line width given the following probability density function for excess line width:

Sol'n:      By inspecting the plot of f(x), below, we might estimate that the expected value or mean or μ for f(x) is approximately 1/3.

To find the exact value of E(x), (i.e., μ), we use the formula that defines it:

Substituting for f(x), and observing that f(x) is nonzero only between 0 and 1, we have

.

After integrating, we have

.