Binomial Distribution: Stochastic Dominance Suppose

*

*$X_1 \sim \operatorname{Bin}(N_1,p)$ and $X_2 \sim \operatorname{Bin}(N_2,p)$

*$N_2>N_1$
Does $X_2$ first-order stochastically dominate $X_1$?


Lemma: If $X,Y$ are random variables on a common probability space where $X\ge Y$ holds always, then $X$ scholastically dominates $Y$.

Proof: $\{Y>c\}\subseteq \{X>c\}$, so $P(Y>c)<P(X<c)$. $\square$
Now, flip $N_2$ independent coins where each has a probability $p$ of heads. Let $X_2$ be the total number of heads, and let $X_1$ be the total number of heads in a particular subset of $N_1$ coins. Clearly $X_2\ge X_1$, so $X_2$ stochastically  dominates $X_1$. 
