Binomial Distribution, Stochastic Dominance, Fixed Mean, etc.
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From: YQQ on 4 Jul 2010 03:48 Hi all, My short question is the following: Fix a natural number mu>0. Let X be a binomial random variable with parameters n and mu/n. So the expectation is mu. Prove that if k<=mu, then Pr[X>=k] decreases as n increases. I verified this for k,n up to 100 using matlab. But proving it seems nontrivial. So pointers to references are greatly appreciated! Qiqi
From: Henry on 4 Jul 2010 17:54
On 04/07/2010 08:48, YQQ wrote: > Hi all, > > My short question is the following: > > Fix a natural number mu>0. Let X be a binomial random variable with > parameters n and mu/n. So the expectation is mu. > > Prove that if k<=mu, then Pr[X>=k] decreases as n increases. > > I verified this for k,n up to 100 using matlab. But proving it seems > nontrivial. It is what you might expect as the mean by construction remains mu, while the variance is mu*(1-mu/n), so increases with increasing n, with the probability in each tail growing as it converges in distribution towards a Poisson distribution. |