Question

the moment generating function of bernoulli distribution is =1-p+pe^t. use this to calculate the mean and variance of the distribution

Please try to explain the solution in words also

Answer 1

We know that given a moment generating function (mfg)   of a random variable X,

then the nth moment is equal to the nth derivative of evaluated at t=0. That is

In our case, we know that the mgf of X, where X has a Bernoulli distribution is

The expectation of X, E(X), is the mean of X. The expectation of X is (that is n=1 above) got by taking the first derivative of mgf and evaluate it at t=0

First we get the first derivative of the mgf

Now we evaluate this at t=0 and get the expectation

The mean of X is p

Next we find the expectation of . First we take the second derivative of the mgf

Now we evaluate this at t=0 and get

Finally we use the formula for the variance to get

The mean of X is

the variance of X is