# Finding the Moment Generating Function of the Geometric Distribution

• Aug 7th 2009, 02:48 PM
bigdoggy
Finding the Moment Generating Function of the Geometric Distribution
Hi
Not too sure if this is the correct area, but hopefully someone can help!
I've got a problem that has arisen from another problem:
\displaystyle \begin{aligned}\sum_{k=0}^\infty e^{tk} p(1-p)^k &=pe^t \sum_{k=0}^\infty (e^t(1-p))^k \end{aligned}

but I can't see how to go from the last line to:

$\displaystyle \begin{array}{cc}p(1-p)^{k-1}&\mbox{k=1,2,3...} \end{array}\\$
• Aug 7th 2009, 03:22 PM
Plato
Quote:

Originally Posted by bigdoggy
Not too sure if this is the correct area, but hopefully someone can help!
I've got a problem that has arisen from another problem:
\displaystyle \begin{aligned}\sum_{k=0}^\infty e^{tk} p(1-p)^k &=pe^t \sum_{k=0}^\infty (e^t(1-p))^k \end{aligned}
but I can't see how to go from the last line to:
$\displaystyle \begin{array}{cc}p(1-p)^{k-1}&\mbox{k=1,2,3...} \end{array}\\$

Frankly I see multiple problems with this question.
Maybe you should post the original question.

This bit of mathematical factoring is impossible.
\displaystyle \begin{aligned}\sum_{k=0}^\infty e^{tk} p(1-p)^k &=\color{red}pe^t \sum_{k=0}^\infty (e^t(1-p))^k \end{aligned}
• Aug 7th 2009, 04:41 PM
bigdoggy
Hi

Yes you're right!

\displaystyle \begin{aligned}\sum_{k=1}^\infty e^{tk} p(1-p)^{k-1 } &= \frac{p}{1-p} \sum_{k=0}^\infty (e^t(1-p))^k \end{aligned}

How would I solve this to produce:

$\displaystyle \frac{pe^t}{1-(1-p)e^t}$
• Aug 7th 2009, 10:55 PM
mr fantastic
Quote:

Originally Posted by bigdoggy
Hi

Yes you're right!

\displaystyle \begin{aligned}\sum_{k=1}^\infty e^{tk} p(1-p)^{k-1 } &= \frac{p}{1-p} \sum_{k=0}^\infty (e^t(1-p))^k \end{aligned}

How would I solve this to produce:

$\displaystyle \frac{pe^t}{1-(1-p)e^t}$

It would help if you posted the original question (as already requested) which I assume was finding the moment generating function of the geometric distribution.

You will find that that particular question has been solved at least once in these forums (use the Search tool).
• Aug 7th 2009, 11:15 PM
Moo
I wonder what's going on with MGF... there have been several these last 2 days (Surprised)

And this question is part of this : http://www.mathhelpforum.com/math-he...97251-mgf.html

If you didn't understand some steps, you could've just asked them there, I would have answered...
• Aug 7th 2009, 11:22 PM
mr fantastic
Quote:

Originally Posted by Moo
I wonder what's going on with MGF... there have been several these last 2 days (Surprised)

And this question is part of this : http://www.mathhelpforum.com/math-he...97251-mgf.html

If you didn't understand some steps, you could've just asked them there, I would have answered...

Therefore ..... thread closed.