= Solution
For a <CUSUM> threshold $h$, define the first signalling time
$$
\tau_h=\inf\{t\geq1:X_t\geq h\}.
$$
The <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 $\operatorname{ARL}_0$ is computed under the <null hypothesis>; a large value means false signals occur infrequently. The out-of-control value $\operatorname{ARL}_A$ 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.
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