We say that a random variable $X$ stochastically dominates a random variable $Y$ if $P(X\ge t)\ge P(Y\ge t).$
I wonder if it is true that for two binomial random variables $X=Bin(n_1,p), Y=Bin(n_2,p),$ $X$ dominates $Y$ if $n_1\ge n_2?$
Actually it is known that one binomial random variable dominates another if the total trials are same but success probabilities are different. How to show a binomial random variable dominates another binomial random variable with a smaller success value?
It seems we only need to verify all natural numbers $t.$ For $t\in [n_2,n_1],$ $P(X\ge t)\ge P(Y\ge t)=0$ is ture. For smaller $t,$ it involves many binomial coefficients. I don't find an easy way to show it.