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 Advanced Statistics Advanced Probability and Statistics Math Forum

 March 19th, 2017, 08:45 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, 09:59 AM #2 Senior Member   Joined: Sep 2015 From: USA Posts: 2,531 Thanks: 1390 you can differentiate it, I'm using $k$ instead of $x$ as is common for discrete values. $P[k] = \binom{n}{k} \theta^k (1-\theta)^{n-k}$ $\dfrac{\partial}{\partial \theta} P[k] = \binom{n}{k}\left(k \theta^{k-1}(1-\theta)^{n-k} - \theta^k(n-k)(1-\theta)^{n-k-1}\right) =$ To find the maximum we set this equal to zero as usual to obtain $k \theta^{k-1}(1-\theta)^{n-k} = \theta^k(n-k)(1-\theta)^{n-k-1}$ $k(1-\theta) = \theta (n-k)$ $k = n\theta$ $\theta = \dfrac {k}{n}$ 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 $P=1$, 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, 03:31 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. Tags binomial, maximum Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post beesee Probability and Statistics 5 September 18th, 2015 01:38 PM Mcfok Trigonometry 2 July 15th, 2015 12:46 PM Monox D. I-Fly Pre-Calculus 4 October 13th, 2014 05:58 AM MATHEMATICIAN Calculus 4 April 19th, 2014 09:20 AM servant119b Algebra 2 June 7th, 2012 02:27 PM

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