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Let $X \sim \text{Bin}(n,p)$ and $Y \sim \text{Bin}(n-1,p)$ with $n >1, p \geq 1/2$ and $X,Y$ are independent. I'd like to show $$(X\mid X \geq 1) \succeq_{sd} 1 + Y.$$ Here $(X \mid \cdot)$ is the conditioned random variable and $\succeq_{sd}$ denotes stochastic dominance, formally: $$\mathbf P[X \geq k \mid X \geq 1] \geq \mathbf P[1+ Y \geq k], \qquad \forall k\geq 1.$$

The intuition is that with $X$ conditioned to be larger than 1 we are guaranteed to have one success. Moreover, with $p \geq 1/2$ the conditioning only makes the remaining $n-1$ Bernouli r.v.s in $X$ more likely to be 1. So, they ought to dominate an independent $\text{Bin}(n-1,p)$.

Note this isn't true in the case $n=2$ and $p=\epsilon$ for small enough $\epsilon$.

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This does not seem to be true. Take $p=2/3$ and $n=k=1000$. Then, you need that $$ \frac{(2/3)^{1000}}{1-(1/3)^{1000}} \geq (2/3)^{999}, $$ which is clearly false.

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  • $\begingroup$ Yes. Thanks. This would have been a nice sufficient condition for a lemma I need. Interesting that you answered, the lemma concerns showing recurrence for the frog model on d-ary trees with Poi(Cd) frogs per-site. $\endgroup$ – Matthew Junge Nov 14 '15 at 19:18
  • $\begingroup$ Well, good luck! Also, consider marking this question as answered then, I want to boost my reputation on this site :) $\endgroup$ – Serguei Popov Nov 14 '15 at 19:27

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