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### Maximum Value of Binomial Random Variables

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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/
```
Associated Topics:
College Statistics

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