# Randomly Stopped Sums vs the sum of I.I.D. Random Variables

• I
• CGandC
In summary, there are two theorems related to Probability Generating Functions. The first one states that the PGF of the sum of independent random variables is equal to the product of their individual PGFs. The second theorem involves a sequence of i.i.d. random variables and a random variable N, which determines the number of terms in the sum. The PGF of the sum in this case is given by the PGF of N multiplied by the PGF of the individual random variables. However, in order to calculate the expected value of t^Y, we need to use conditional expectation to remove the dependence on N.

#### CGandC

I've came across the two following theorems in my studies of Probability Generating Functions:

Theorem 1:
Suppose ##X_1, ... , X_n## are independent random variables, and let ##Y = X_1 + ... + X_n##. Then,
##G_Y(s) = \prod_{i=1}^n G_{X_i}(s)##

Theorem 2:
Let ##X_1, X_2, ...## be a sequence of independent and identically distributed random variables with common PGF ##G_X##. Let ##N## be a random variable, independent of the ##X_i##'s with PGF ##G_N##, and let ##T_N = X_1 + ... + X_N = \sum_{i=1}^N X_i##. Then the PGF of ##T_N## is:
##G_{T_N}(s) = G_N (G_X(s))##

Question:
I don't understand the difference between these two theorems.
I understand that in first theorem ## n ## is a number that we know so we know how many ## X_i ## will appear in the sum in ## Y ##.
But in the second theorem ## N ## is a random variable so we don't know how many ## X_i ## will appear in the sum ## Y ##.

But I still don't fully understand.

the proof for the first theorem goes as follows:
##
G_Y(t) =G_{X_1+X_2+\ldots+X_n}(t)=\mathbb{E}\left[t^{X_1+X_2+\ldots+X_n}\right]=\mathbb{E}\left[\prod_{i=1}^n t^{X_i}\right]=\prod_{i=1}^n \mathbb{E}\left[t^{X_i}\right]=\prod_{i=1}^n G_{X_i}(t)
##

Then I tried to prove the second theorem using exactly the same proof as follows:
##
G_Y(t) =G_{X_1+X_2+\ldots+X_N}(t)=\mathbb{E}\left[t^{X_1+X_2+\ldots+X_N}\right]=\mathbb{E}\left[\prod_{i=1}^N t^{X_i}\right]=\prod_{i=1}^N \mathbb{E}\left[t^{X_i}\right]=\prod_{i=1}^N G_{X_i}(t)
##
this proof is specious, but I don't understand why. I mean, the number of ## X_i## 's that will be multiplied by each other is determined by ## N ## ,even if we don't know it, so I don't understand what's the problem.

Thanks in advance for any help!

CGandC said:
I've came across the two following theorems in my studies of Probability Generating Functions:

Theorem 1:
Suppose ##X_1, ... , X_n## are independent random variables, and let ##Y = X_1 + ... + X_n##. Then,
##G_Y(s) = \prod_{i=1}^n G_{X_i}(s)##

Theorem 2:
Let ##X_1, X_2, ...## be a sequence of independent and identically distributed random variables with common PGF ##G_X##. Let ##N## be a random variable, independent of the ##X_i##'s with PGF ##G_N##, and let ##T_N = X_1 + ... + X_N = \sum_{i=1}^N X_i##. Then the PGF of ##T_N## is:
##G_{T_N}(s) = G_N (G_X(s))##

Question:
I don't understand the difference between these two theorems.
I understand that in first theorem ## n ## is a number that we know so we know how many ## X_i ## will appear in the sum in ## Y ##.
But in the second theorem ## N ## is a random variable so we don't know how many ## X_i ## will appear in the sum ## Y ##.

But I still don't fully understand.

the proof for the first theorem goes as follows:
##
G_Y(t) =G_{X_1+X_2+\ldots+X_n}(t)=\mathbb{E}\left[t^{X_1+X_2+\ldots+X_n}\right]=\mathbb{E}\left[\prod_{i=1}^n t^{X_i}\right]=\prod_{i=1}^n \mathbb{E}\left[t^{X_i}\right]=\prod_{i=1}^n G_{X_i}(t)
##

Then I tried to prove the second theorem using exactly the same proof as follows:
##
G_Y(t) =G_{X_1+X_2+\ldots+X_N}(t)=\mathbb{E}\left[t^{X_1+X_2+\ldots+X_N}\right]=\mathbb{E}\left[\prod_{i=1}^N t^{X_i}\right]=\prod_{i=1}^N \mathbb{E}\left[t^{X_i}\right]=\prod_{i=1}^N G_{X_i}(t)
##
this proof is specious, but I don't understand why. I mean, the number of ## X_i## 's that will be multiplied by each other is determined by ## N ## ,even if we don't know it, so I don't understand what's the problem.

Thanks in advance for any help!

$\prod_{i=1}^N G_{X_i}(t) = (G_{X_1}(t))^N$ is a random variable: it's a function of $N$. To find $\mathbb{E}(t^Y)$ you need to remove this dependence on $N$ by using conditional expectation: $$\begin{split} \mathbb{E}(t^{Y}) &= \sum_{n=1}^\infty \mathbb{E}(t^{X_1 + \dots + X_N} | N = n)\mathbb{P}(N = n) \\ &= \sum_{n=1}^\infty \mathbb{E}(t^{X_1 + \dots + X_n})\mathbb{P}(N = n) \end{split}$$

• jim mcnamara and CGandC
Ahh! that makes sense, thank you alot!

• jim mcnamara