It is very straightforward to prove $X\sim Binomial(n+1,\delta)$ first-order stochastically dominate $W\sim Binomial(n,\delta)$. However, is it also true that $X\sim B(Y1,\delta)$ first-order stochastically dominate(FSD) $W\sim B(Y0,\delta)$ when $Y1$ first-order stochastically dominate $Y0$? That is, in the latter case, $Y1$ and $Y0$ are random variables.
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Obviously yes, simply select some random variables $Z_i\sim Y_i$ such that $Z_1\geqslant Z_0$ almost surely and compare Binomial$(Z_1,\delta)$ and Binomial$(Z_0,\delta)$. – Did Sep 27 '18 at 17:50
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@Did Thank you. But I have some difficulty in proving this little bit more formally. – user3509199 Sep 27 '18 at 17:56
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Which part? Please be specific. – Did Sep 27 '18 at 18:00
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@Did How can I compare Binomial($Z_1,\delta$) with Binomial($Z_0,\delta$) where $Z_1>Z_0$ almost surely. – user3509199 Sep 27 '18 at 18:04
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Well, realize every Bin$(n,\delta)$ simultaneously as $\sum\limits_{k=1}^nU_k$ from a single i.i.d. Bernoulli sequence $(U_k)$ independent of $(Z_0,Z_1)$, then $\sum\limits_{k=1}^{Z_1}U_k\geqslant\sum\limits_{k=1}^{Z_0}U_k$ with full probability. – Did Sep 27 '18 at 18:05
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@Did Thank you very much. – user3509199 Sep 27 '18 at 18:08