= Solution
Interpret each mechanism for a fixed missingness pattern $r$: the observed and missing response components are the coordinates selected by that pattern, and the <covariates> $X$ are assumed fully recorded.
For <missing completely at random>, missingness depends on neither response values nor <covariates>:
$$
\boxed{f(r\mid Y^o,Y^m,X)=f(r).}
$$
For <covariate-dependent missing completely at random>, response observation may depend on $X$ but not on the response after conditioning on $X$:
$$
\boxed{f(r\mid Y^o,Y^m,X)=f(r\mid X).}
$$
This is <conditional independence> $R\perp Y\mid X$ and can still produce a marginal association between response and missingness.
For <covariate-dependent missing at random>, the observation probability may use observed response components and <covariates>, but not the values missing under that pattern:
$$
\boxed{f(r\mid Y^o,Y^m,X)=f(r\mid Y^o,X).}
$$
For <missing not at random>, there remains dependence on $Y^m$ after conditioning on $Y^o,X$; the displayed simplification is not valid. For example, refusal could depend on the unreported response itself, even among individuals sharing all recorded <covariates>. These mechanisms concern the reason for nonobservation, not whether the incomplete records merely look irregular. Their distinction is not generally identifiable from the observed responses alone.
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