
Advanced Statistics Advanced Probability and Statistics Math Forum 
 LinkBack  Thread Tools  Display Modes 
March 19th, 2017, 08:45 AM  #1 
Newbie Joined: Nov 2013 Posts: 25 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: CA Posts: 1,111 Thanks: 580 
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, 03:31 PM  #3 
Newbie Joined: Nov 2013 Posts: 25 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  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
How to resolve binomial experiment without binomial theorem.  beesee  Probability and Statistics  5  September 18th, 2015 01:38 PM 
Maximum value  Mcfok  Trigonometry  2  July 15th, 2015 12:46 PM 
Difference Between Maximum/Minimum Value and Maximum/Minimum Turning Point  Monox D. IFly  PreCalculus  4  October 13th, 2014 05:58 AM 
maximum value  MATHEMATICIAN  Calculus  4  April 19th, 2014 09:20 AM 
A = Maximum(AB, AC, CB)  servant119b  Algebra  2  June 7th, 2012 02:27 PM 