Probability Generating Functions

[EthanW]

Hello,

I am trying to get the hang of Probability Generating Functions, but I don't quite understand them fully.

For example, I've got the PGF of a random variable X, called H:
$$H(s) = \frac{1}{3}\cdot(1+s+s^2)$$

Now, then there is a random variable Y, with Y = X + 1, and I want to solve the PGF of Y I do:
$$G_Y(s) = G_{x+1}(s) = E[s^{x+1}] = E[s^x]\cdot E<s> = ? </s>$$

I don't know how to go further at this point, can someone point me in the right direction?

Thanks.

 

[awkward]

Hello,

I am trying to get the hang of Probability Generating Functions, but I don't quite understand them fully.

For example, I've got the PGF of a random variable X, called H:
$$H(s) = \frac{1}{3}\cdot(1+s+s^2)$$

Now, then there is a random variable Y, with Y = X + 1, and I want to solve the PGF of Y I do:
$$G_Y(s) = G_{x+1}(s) = E[s^{x+1}] = E[s^x]\cdot E<s> = ? </s>$$

I don't know how to go further at this point, can someone point me in the right direction?

Thanks.

Your original PGF simply means that P(0) = P(1) = P(2) = 1/3.

If you want P(1) = P(2) = P(3) = 1/3, the generating function is

$$\frac{1}{3} (s + s^2 + s^3)$$

 

[EthanW]

Thanks, very useful information. I made some other exercises and they've become more clear now. :)