MGF and the binomial distribution
The moment generating function M(t), can give us the different moment of a distribution and it’s sometimes more easy to compute mean and variance this way.
I propose to show how to do it with the binomial distribution. We can describe the binomial distribution this way:
This distribution describe the probability of an event to occure x times in n tests. It’s not so useful in physics, but it help to introduce the Poisson and normal distribution which are very important in physics and mathematics.
If we compute the MGF we get this result:
This is not a function to complicated to derive and that’s fun, because we are going to devide it two times.
To get the nth moment of the binomial distribution, we only need to let t=0 in the nth derivative of the MGF. Wcan compute the mean just to be sure this is true.
This is useful and so why not trying to compute the variance this way?
This is a useful function and as we see it give good results really fast. Even if I didn’t show as a proof that it work, we can still appreciate how it work his case. If you want to see a proof of this function I’m probably sure google can help.
My source for this was the book A first course in probability of Sheldon M. Ross or in french, Initiation aux probabilités.
