This question might be a bit too specific but an answer will help me grasp if I understood the concept of random processes correctly. As we may know, a random process $X(t,w)$ where the continuous variable chosen here is time and $w$ is the set of all possible outcomes of my random variable, can be defined from two perspectives:

-As a collection of random variables after fixing the time $t = t_1$ for example, for each outcome $w$

-As a collection of sample functions, where we get continuous functions in t for each outcome $w$

So far my understanding is that, if I take different time instants and compute the ensemble average for each collection of RVs, I would use the same density function for each computation. My question is the following: is it possible to have for a random process at different time instants, assuming the same set of possible outcomes, different probability density functions (i.e at $t= t_1$ it's Gaussian, $t= t_2$ it's uniform, etc.)? Does this imply that my pdf is time varying? In other words, for a "regular" random process, is the PDF always the same but only it's statistics are time varying? Or is there a flaw in my understanding of this concept? I hope this question is clear and please do let me know if this is not the place to post this question, this is the first time on stackExchange!


I'm not entirely sure I understand what you're asking, but there are not really any restrictions on the PDFs of stochastic processes. We can have any distribution for $X_{t_1}$ and any other (or the same) distribution for $X_{t_2}$ (as long as $t_1 \ne t_2$, of course). In general, processes that have the same distribution over time come up less often than processes whose distribution changes over time, i.e. there is no reason to expect that $X_{t_1}$ and $X_{t_2}$ have the same distribution.


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