  We give a stochastic comparison and ordering of the largest eigenvalues, with
parameter $\beta$, for Hermite $\beta$-ensembles and Laguerre
$\beta$-ensembles. Taking limit, we recover a stochastic domination result for
Tracy-Widom distributions obtained in \cite{VP2022}. We prove a stochastic
domination result for the higher order analogues of Tracy-Widom distributions.
We also generalize a few stochastic domination results from last passage
percolation.
