Maximum Value of Binomial Random Variables

Date: 09/12/2001 at 12:03:15
From: Clark Thorne
Subject: Maximum Value of Binomial Random Variables

For a binomial random variable X with parameters (n,p), show that 
P{X=i} first increases and then decreases, reaching its maximum value
when i is the largest integer less than or equal to (n+1)p.

I first tried to differentiate the binomial distribution, but fell on 
my face with the gamma function. This seems intuitive, but how do you 
show this for generic parameters (n,p)?

Date: 09/12/2001 at 15:38:00
From: Doctor Jubal
Subject: Re: Maximum Value of Binomial Random Variables

Hi Clark,

Thanks for writing to Dr. Math.

Based on your question, I'll assume you know that for a binomial 
distribution, the probability that X=i is given by

                                   n!
  P{X=i} = p^i * (1-p)^(n-i) * ----------
                                i!(n-i)!

where n is the number of trials and p is the probability of some 
outcome on a given trial. I assume you tried to differentiate this, 
and found it hard. Let me suggest a different way.

Let's compare the probability that X=i to the probability that X=i+1.  
If P{X=i}/P(X=i+1) < 1, then the distribution function is increasing.  
If the ratio is greater than one, the distribution function is 
decreasing. For the binomial distribution

                                             n!
                   i          (n-i)      ----------
    P{X=i}        p    *  (1-p)      *    i!(n-i)!
  ---------- = -----------------------------------------
   P{X=i+1}      (i+1)       (n-i-1)          n!
                p      * (1-p)       * ----------------
                                        (i+1)!(n-i-1)!

By canceling terms that appear in both numerator and denominator, we 
can simplify this to

    P{X=i}      (1-p) * (i+1)!(n-i-1)!     (1-p)(i+1)
  ---------- = ------------------------ = ------------
   P{X=i+1}       p   *   i!(n-i)!           p(n-i)

Since the distribution function is increasing when this ratio is less 
than one, the binomial function is increasing when

      p(n-i) > (1-p)(i+1)

     np - ip > i + 1 - ip - p

          np > i + 1 - p

      np - p > i + 1

      (n+1)p > i + 1

Similarly, the binomial function is decreasing when i + 1 > (n+1)p.

This is exactly what you were trying to prove.  So long as i+1 is less 
than (n+1), P{X=i+1} is greater than P{X=i}, and the distribution 
function is increasing. But if i is the largest integer less than or 
equal to (n+1)p, then i+1 is greater than (n+1)p, and the distribution 
function is decreasing for this and all greater values of i. 
Therefore, the binomial function is increasing for small i, decreasing 
for large i, and has a single maximum at i = floor[ (n+1)p ]. 

Does this help?  Write back if you'd like to talk about this some
more, or if you have any other questions.

- Doctor Jubal, The Math Forum
  http://mathforum.org/dr.math/