msv-40 - The Task and Answers

MSV 40:
The Binomial Mean and Variance
One of the first things you
ask on being shown a new
probability distribution is:
‘What’s the mean? And what’s the variance?’
For the Binomial Distribution, there are simple answers.
If X ~ B(n, p), then the mean of X = E(X) = np.
If X ~ B(n, p), then the variance of X = Var(X) = npq,
where q = 1 - p.
Being a good mathematician, you will ask,
‘Can I prove these things?’
E(X) = np makes sense.
If you roll a normal dice 60
times, then you would expect on
average the number of sixes
rolled to be
10 = 60 x 1/6.
The formula for the variance is harder to see.
Let’s try to prove these results
for a special case.
Suppose X ~ B(3, p) for some p.
Can you prove that E(X) = 3p
and Var (X) = 3p(1 - p) ?
If X ~ B(3, p) then we have
= 3p.
Var(X) = E(X2) - (E(X))2.
So Var(X) = E(X2) - (E(X))2 = 6p2 + 3p – (3p)2
= 3p – 3p2 = 3p(1-p).
With thanks to
is written by Jonny Griffiths