MSV 40: The Binomial Mean and Variance www.making-statistics-vital.co.uk MSV 40: The Binomial Mean and Variance

‘What’s the mean? And what’s the 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.

‘Can I prove these things?’ 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) ? Try!

If X ~ B(3, p) then we have = 3p.

So Var(X) = E(X2) - (E(X))2 = 6p2 + 3p – (3p)2 = 3p – 3p2 = 3p(1-p).

is written by Jonny Griffiths With thanks to pixabay.com www.making-statistics-vital.co.uk is written by Jonny Griffiths hello@jonny-griffiths.net