# Homework Help: Probability function

1. Apr 23, 2014

### JoanneTan

Here is the question,

f(x) = ce^-x , x = 1, 2, 3.....

Find the value of c.
Find the moment generating function of X.
Use the result obtained, find E(x).
Find the probability generating function of X.
Verify that E(x) obtained using probability generating function is same as the first E(x) founded.

I check the answer of the book, but it's wrong. Can someone help me? I'm looking for the working.

2. Apr 23, 2014

### Ray Vickson

PF rules require that you show your work.

3. Apr 23, 2014

### JoanneTan

Ok.. For the first question which is to find value of c.

But the answer given is e - 1.
It's not e^-1.. I'm confusing how to get e - 1.

4. Apr 23, 2014

### HallsofIvy

Yes, you must have $ce^{-1}+ ce^{-2}+ ce^{-3}+ \cdot\cdot\cdot= 1$

You can factor out $ce^{-1}$ and have $ce^{-1}(1+ e^{-1}+ e^{-2}+ \cdot\cdot\cdot)$

That is, as you say, a geometric series with common factor $e^{-1}$ so is equal to $\frac{ce^{-1}}{1- e^{-1}}= 1$. That, you have. Now multiply both numerator and denominator by $e$:
$\frac{c}{e- 1}= 1$.

Last edited by a moderator: Apr 23, 2014
5. Apr 23, 2014

### JoanneTan

Oh! Ok.. I get u now.. Thanks a lot! But the second question, moment generating function,
As u can see from the photo, I done until half, can help me to continue? Cause dono how to substitute.

6. Apr 23, 2014

### Ray Vickson

You need to calculate the sum $$\sum_{n=1}^{\infty} e^{-n} e^{kn} = \sum_{n=1}^{\infty} r^n, \text{ where } r = e^{k-1}$$
You have already seen how to do such summations; look at part (a)!