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March 19th, 2017, 09:45 AM  #1 
Newbie Joined: Nov 2013 Posts: 29 Thanks: 1  Maximum of a binomial
If X is a binomial random variable, for what value of theta is the probability b(x;n,theta) a maximum? In their notation theta is the probability of a success and n is the number of trials. I'm not sure where to start to be honest. For a given value of the parameters I could say the maximum occurs around mean and do some algebra to find it. But with a parameter it seems somewhere different. Plus, how would a curve be bigger than another curve? If a curve is greater than another curve for all possible values then it should have a greater sum when integrated over the support but these curves have the same area since they're probabilistic. So it leaves me to think that I need to compare the maximums associated with each given curve. To find the maximums of the maximums. If so, would the answer be 1? Because mean=n*theta and theta is in the interval [0,1]. If only this was a continuous function, I'd be able to differentiate this thing and move on. I appreciate anyones feed back. 
March 19th, 2017, 10:59 AM  #2 
Senior Member Joined: Sep 2015 From: USA Posts: 1,656 Thanks: 842 
you can differentiate it, I'm using $k$ instead of $x$ as is common for discrete values. $P[k] = \binom{n}{k} \theta^k (1\theta)^{nk}$ $\dfrac{\partial}{\partial \theta} P[k] = \binom{n}{k}\left(k \theta^{k1}(1\theta)^{nk}  \theta^k(nk)(1\theta)^{nk1}\right) =$ To find the maximum we set this equal to zero as usual to obtain $k \theta^{k1}(1\theta)^{nk} = \theta^k(nk)(1\theta)^{nk1}$ $k(1\theta) = \theta (nk)$ $k = n\theta$ $\theta = \dfrac {k}{n}$ Now, we note that the maximum value on the right hand side occurs at $k=n$ because $0 \leq k \leq n$ Thus across the entire distribution the maximum of $P[k]$ occurs when $\theta = 1$, and $P[n]=1$ We can note that the distribution is just mirrored when $\theta = 0$ with $P[0]=1$, this is also a maximum. So in order to maximize a binomial distribution by adjusting $\theta$ you select $\theta=0$ or $\theta=1$ to obtain a maximum value of 1. 
March 19th, 2017, 04:31 PM  #3 
Newbie Joined: Nov 2013 Posts: 29 Thanks: 1 
Thanks! I didn't think it was valid to differentiate a discrete function but in hindsight I can reason why its okay.


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