Thread: moment-generating function

We went over this material once in class and we have a bunch of problems that we have to do, and I still don't know if I understand the concept fully. One of the problems is:

Find the moment-generating function of the discrete random variable X that has the probability distribution
f(x) = 2(1/3)^x for x = 1, 2, 3, ...
and use it to determine the values of mu1' and mu2'.

So I tried to use the definition, which says that Mx(t) = E(e^(tx)) = Integral from -INF to INF of (e^(tx))*f(x)dx.

I got the Integral from 1 to INF of (e^(tx))*(2(1/3)^x)dx, but now I am lost... any help would be GREATLY appreciated.

The moment generating function definition you have used is for a continuous random variable with probability distribution $\displaystyle f(x)$.

In every case we can use the definition: $\displaystyle m_X(t) = E[e^{tX}] $.

So in your case:
$\displaystyle m_X(t) = 2 \left(\frac{1}{3}e^t + \left(\frac{1}{3}\right)^2 e^{2t} + ...\right)$$\displaystyle = \frac{2}{3}e^{t} \left(1+\frac{1}{3}e^t + \left(\frac{1}{3}e^t \right)^2 + \left(\frac{1}{3}e^t\right)^3+... \right)$
$\displaystyle =\frac{\frac{2}{3}e^t}{1-\frac{1}{3}e^t} = \frac{2e^t}{3-e^t}$

I see your first two steps, but how did your third step become so simplified? Could you show me the step in between that? Thanks!

This is the geometric series:

$\displaystyle \sum_{i=0}^\infty ar^{i} = \frac{a}{1-r}$

Thank you so much! That really helps a lot. Now I will see if I can do the rest on my own.

Also, could I ask, anyone, what program you have on your computer that lets you type mathematically?

Originally Posted by TheHolly

Also, could I ask, anyone, what program you have on your computer that lets you type mathematically?

It's called Latex and it isn't a program. It's built right into this site so any user can use it.

You see the button beside the YouTube button when you make a post? That button wraps the selected text with [tex] tags.

You can also click on the math equations you see on here to find out how they were written. For example, to write:

$\displaystyle
\sin {\sqrt {4x^2}} = \log {\frac {1}{2}}$

I would have to write:

\sin {\sqrt {4x^2}} = \log {\frac {1}{2}}

And then wrap it with math tags.

Sorry, guys, but I still can't figure out mu1' or mu2'. Can anyone help???

Originally Posted by TheHolly

Sorry, guys, but I still can't figure out mu1' or mu2'. Can anyone help???

By mu1' do you mean E(X)?
By mu2' do you mean E(X^2)?

If so, note that


$\displaystyle E(X^n) = \left[ \frac{d^n M_X}{dt^n} \right]_{t = 0}$.


that is, evaluate the nth derivative of $\displaystyle M_X$ at t = 0.

Solve for n = 1 (I get 3/2) and n = 2 (I get 3).