The Pilsdorff Beer Company runs a fleet of trucks along the $100$ mile road from Hangtown to Dry Gulch. The trucks are old, and are apt to break down at any point along the road with equal probability. Where should the company locate a garage so as to minimize the expected distance from a typical breakdown to the garage? In other words, if $X$ is a random variable giving the location of the breakdown, measured, say, from Hangtown, and $b$ gives the location of the garage, what choice of $b$ minimizes $E(|X − b|)$?
Now, suppose $X$ is not distributed uniformly over $[0, 100]$, but instead has density function $f_X(x) = \frac{2x}{10000}$. Then what choice of $b$ minimizes $E(|X − b|)$?
Attempt: For this, I thought about taking the definite integral $$\int_{0}^{100}(2x/10,000)dx,$$ and got $1$, but it did not seem to be leading to the right answer. Later, I was really stuck.
