Assume that $X$ is Bin$(n, p)$ and that $Y$ is Bin$(m, q)$. Under which conditions on $(n, m, p, q)$ does the property that $X \preceq Y$ hold?
I know that if $n \leq m$ and $p=q$, then $X \preceq Y$, as can be proved using a coupling argument considering $n$ and $m$ Bernoulli random variables.
But I don't know how to tackle the problem when both $n$ and $p$ can vary.