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*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.

From reading here: https://stats.stackexchange.com/que...topped-sums-vs-the-sum-of-i-i-d-random-variab

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!