 Solution

For a CUSUM threshold $h$, define the first signalling timeThe average run length is $\mathbb{E}({\tau }_h)$ under a specified distribution of the observations and specified initial state, here $X_0=0$. The in-control value ${\mathrm{ARL}}_0$ is computed under the null hypothesis; a large value means false signals occur infrequently. The out-of-control value ${\mathrm{ARL}}_A$ is computed under a specified alternative hypothesis; a small value means rapid detection when that alternative holds from monitoring's start. If a change occurs later, its detection delay depends also on the chart state at change time, so the zero-state alternative average run length is not every possible delay.

A fixed-horizon statistical hypothesis test is characterized by Type I error probability under its null and Type II error probability under a specified alternative. Continuous monitoring repeatedly offers opportunities to signal: over an indefinitely long run its probability of eventually signalling can be one even under the null. The average run length quantifies waiting time rather than that eventual probability. It replaces the single-test emphasis on a false-rejection probability with a false-signal timescale, and compares detection speed rather than only a fixed-horizon miss probability. However, a mean run length alone does not determine either finite-horizon error probability; those require the run-length distribution.