Let $X_1, X_2$ and $Y$ be nonnegative random variables, and $M=max(X_1, Y)$ and $N=max(X_2, Y)$. The CDFs of each variable are $F_{X_1}, F_{X_2}, F_{Y}, F_{M},$ and $F_{N}$.
$Proposition$:
$$E[M]\geq{}E[N]$$ if and only if $$F_{X_1}(t)\geq{}F_{X_2}(t)$$ for $t\geq{}0$
$Proof$:
CDF of $M$ is $$F_M(m)=F_{X_1}(m)F_Y(m)$$ since $M=max(X_1, Y)<m$ if and only if $X_1<m$ and $Y<m$. Similarly, CDF of $N$ is $$F_N(n)=F_{X_2}(n)F_Y(n).$$ Using the fact that $E[X]=\int_0^\infty (1-F_X(t))dt$ for nonnegative random variable $X$ with its CDF $F_x$, $$E[M]\geq{}E[N] \\ \Leftrightarrow{}\int_0^\infty (1-F_M(t))dt \geq{} \int_0^\infty (1-F_N(t))dt \\ \Leftrightarrow{}\int_0^\infty F_N(t)-F_M(t)dt\geq{}0 \\ \Leftrightarrow{}\int_0^\infty F_{X_2}(t)F_Y(t)-F_{X_1}(t)F_Y(t)dt\geq{}0 \\ \Leftrightarrow{}\int_0^\infty F_Y(t)[F_{X_2}(t)-F_{X_1}(t)]dt\geq{}0$$ which is true when $F_{X_1}(t)\geq{}F_{X_2}(t)$ $for$ $t\geq{}0$
Did I proved the proposition correctly? Please give any comments on both of my proposition and proof. References regarding this will be really helpful. Plus, if there is any relaxed condition that satisfies the inequality $E[M]\geq{}E[N]$ please let me know!