Question related to IID process

[Shloa4]

Hi.
I have a question about and IID process (attached). I'll be happy if someone could help me understnad it better.
Thanks in advance :shy:

 

[mathman]

The statement is about the sum. The distribution being the product of individual distributions has nothing to do with sum or product.

 

[chiro]

Hi.
I have a question about and IID process (attached). I'll be happy if someone could help me understnad it better.
Thanks in advance :shy:

Are you familiar with the convolution theorem for sums of IID random variables?

It might be wise to get a book on applied probability and then look at the convolution theorem for the sum of two IID random variables where the author proves that the CDF is the convolution (you can differentiate to get the PDF).

 

[Stephen Tashi]

Shloa4,

If the meaning of the question is clear to you, you should write it out in your own words. I don't think the document is clear.

It says "Let $X_n = X(n) (n = 1,2,3...)$ be an independent identically distributed (IID), discrete-time random process".

I don't understand why there is subscript on $X_n$. I think there is one random process (which I would have called $X$ , with no subscript) and $X_n = X[n]$ is the random variable associated with time $n$. (A "stochastic process" is an indexed collection of random variables.)

Then document defines a process $S_m$ by $S_m = \sum_{n=1}^m X_n$.

Then the document asks for the "common PDF" $f_{X_1,X_2...X_n}(x_1,x_2,..x_n; t_1,t_2,...t_n)$

Is "common PDF" supposed to mean the joint probability density of the vector of random variables $(X_1,X2,...X_n)$ ? If so, I don't see that this question has anything to do with the process $S_m$.

The answer is given as $\sum_{n=1}^m f_{X_n}(x_n,t_n)$

So how did the subscript $m$ get into that formula?