Using the mgf to find the mean and variance.

Question

Let $X$ have a moment-generating function given by

$M(t)= 3/10e^t+1/5e^{2(t)}+1/10e^{(3t)}+2/5e^{(4t)}$

The question is to use the mgr to find the mean and the variance of $X$.

I am not sure how to do these.

Answer 1

The reason why this function is called the moment generating function is that you can obtain the moments of $X$ by taking derivatives of $X$ and evaluating at $t=0$.

$$\left.\frac{d}{dt^n} M(t) \right|_{t=0} = \left.\frac{d}{dt^n} E[e^{tX}] \right|_{t=0} = E[X^n e^{tX}]|_{t=0} = E[X^n].$$

In particular, $E[X]=M'(0)$ and $E[X^2]=M''(0)$. This gives you the mean and the second moment. To obtain the variance, do $Var(X)=E[X^2]-E[X]^2$.