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?

 

[blue_raver22]

Hello,

Thank you for reaching out with your question about the IID process. I would be happy to help you understand it better.

The IID process stands for Independent and Identically Distributed process. It is a statistical concept used to describe a series of random variables that are independent of each other and have the same probability distribution. This means that each variable in the series is not affected by the previous or future variables and follows the same pattern of distribution.

An example of an IID process is flipping a fair coin multiple times. Each flip is independent of the previous one and has a 50% chance of landing on either heads or tails, making it identically distributed.

I hope this helps clarify the concept of IID process for you. If you have any further questions, please don't hesitate to ask. Thank you.