# Show that the variance of a binomial distribution

• Jan 2nd 2010, 05:20 AM
Show that the variance of a binomial distribution
Show that the variance of a binomial distribution with parameters n and p cannot exceed n/4.
• Jan 2nd 2010, 05:54 AM
Moo
1. What is the variance of a binomial distribution (n,p) ?

2. What is the maximum value of the function $\displaystyle f(x)=x(1-x)$, when $\displaystyle 0\leq x\leq 1$ ?
• Jan 3rd 2010, 02:07 AM
Quote:

Originally Posted by Moo
1. What is the variance of a binomial distribution (n,p) ?

2. What is the maximum value of the function $\displaystyle f(x)=x(1-x)$, when $\displaystyle 0\leq x\leq 1$ ?

$\displaystyle \sigma^2 = n \theta(1- \theta)$

and max value of $\displaystyle f(x) = x(1-x) = 0$ when we substitute the value of x= 1
• Jan 3rd 2010, 02:59 AM
mr fantastic
Quote:

Originally Posted by wolfyparadise
$\displaystyle \sigma^2 = n \theta(1- \theta)$

and max value of $\displaystyle f(x) = x(1-x) = 0$ when we substitute the value of x= 1

Maximum value! NOT minimum value. Draw the graph of $\displaystyle y = x(1 - x)$ over the domain $\displaystyle 0 \leq x \leq 1$. (Did you bother to do that before just plucking a random [wrong] answer out of the air?). The maximum value occurs at the turning point. Drawing parabolas and finding turning points is something you should have learned how to do long ago. Do you realise that work previously studied is assumed to be understood?
• Sep 4th 2012, 03:54 PM
tbrauch
Re: Show that the variance of a binomial distribution
I believe that the maximum variance for a binomial distribution occurs when the probability is 1/2 and the minimum variance occurs when probably is 0 or 1. The function of variance is in fact an inverted parabola however the maximal value occurs at 1/2 not 0. A quick check is to plot the variance over different value of p in the range of 0 and 1. A better way is to calculate first derivative = 0 and solve the for p which shows the apex of the parabola --- in this case p is also 1/2.