Example of Stochastic Order I know that stochastic order means the concept of one variable is "bigger" than another. But I would like some example to visualize/understand it better.
 A: I think it is better to think in terms of bets (as the wikipedia article mentions)...bet A: you get a dollar if you roll 4-6 on a die, 0 if you get 1-3. Bet B: you get 1 dollar if you toss heads, 25 cents if you toss tails (unbiased coin). Then your winnings on bet B stochastically dominate winnings on bet A.
A: A pointwise characterization might help: 

$X$ is stochastically dominated by $Y$ if and only if there exists a probability space $(\Omega',\mathcal F',P')$ and two random variables $X'$ and $Y'$ defined on $\Omega'$, such that:
  
  
*
  
*The distributions of $X$ and $X'$ coincide.
  
*The distributions of $Y$ and $Y'$ coincide.
  
*$X'\leqslant Y'$ almost surely, that is, $P'(X'\leqslant Y')=1$.
  

Note that to put back everything on a different probability space $(\Omega',\mathcal F',P')$ before proceeding to pointwise comparisons is, in a way, the best one can expect since $X$ and $Y$ might be defined on different probability spaces.
Equivalently, $X$ is stochastically dominated by $Y$ if and only if there exists $(X',Z')$ such that $X'$ is distributed like $X$, $X'+Z'$ is distributed like $Y$ and $Z'\geqslant0$ almost surely.
