What is a Random Process? 

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Just to be a bit more precise, a sample function doesn't have a pdf. Sometimes people do say that, but it's not an accurate statement. A random variable has a pdf. A sample function can be used to generate a "normalised histogram" over time - and that can be compared to the pdf of the random variables, as I explain starting at the 5:40 min point in the video. If they are the same, then it is called "ergodic".

@@iain_explains is the 'histogram' related to the ensemble average?

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The technical/mathematical definitions are for infinite time waveforms (so there is no "start" or "end"). But in practice we apply these mathematical definitions to finite time duration signals. Also, in practice of course, we only ever have one "realisation" of a signal (unless you believe in infinite "parallel universes").

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There are lots of books that cover these topics. Probably the Schaum series have lots of worked questions (although I haven't used them myself). I like the book: Peyton Z. Peebles Jr, “Probability, Random Variables and Random Signal Principles”

@@iain_explains Thank you for your reply😊

Hopefully this helps: "What does Ergodic mean for Random Processes?" ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-GchHaeVludo.html

The histogram of a sample realisation function/waveform is only the same as the pdf of the stationary random process from which it was sampled, if the process is ergodic. Consider this non-ergodic process: "the speed of runners in the London marathon". If you randomly pick a runner, and plot their speed as a function of time, then take the histogram of that plot (over time), it would not be the same as the overall pdf - because it would depend on which runner you picked. Slow and fast runners would have different histograms. The random process would be stationary (assuming the London marathon is a flat course, and average speeds don't change over time), but not ergodic.

@@iain_explains Thank you!

Great question. It's going to be the topic of one of my upcoming videos. My best "counter example" is to consider the process of the height of planes as they fly over a town in windy turbulent conditions. As each plane flies overhead, its height will go up and down randomly depending on the wind. So the height pdf could be modelled as a Gaussian centred on the height that the flight control tower told them to fly at. But what if there were two allowable heights in the regulations? Then some planes would have a stationary Gaussian pdf centred on one height, and other planes would have a stationary Gaussian pdf centred on the other height. The overall pdf of the overall process of "planes that fly overhead" will be a _stationary_ pdf which is the addition of the two Gaussians. But if you selected a specific plane, and tracked its flight height, and generated a histogram from that time-domain realisation from that plane, then it would only show a single Gaussian. You couldn't learn the overall "two Gaussian" pdf, from any single realisation of the process. Therefore it is not _ergodic_ .

Hopefully this video will help: "What is Rayleigh Fading?" ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE--FOnYBZ7ZfQ.html

@@iain_explains Thanks a lot !!!!!!

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Some answers: 1. This video will help: "What is a Stationary Random Process?" ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-elV4B_-79oo.html 2. Yes, to both of your suggested examples. A general example is any system that changes parameters over time (for example, the maximum daily temperature - it's random, but the PDF changes continually throughout the year). 3. Yes.

@@iain_explains Perfect, thanks a lot!!

A "sample function" is a realisation of a Random Process. In the same way that a "sample" is a realisation of a Random Variable. For example, the outcome of rolling a 6-sided dice is a Random Variable. If you actually roll a dice, and look at the number on the side facing up, then that is a "sample" of that Random Variable.

If the PDFs are the same at all times, then the Random Variables at those times are "identically distributed" (i.d.). For i.i.d. the RVs need to be independent as well.

@@iain_explains Oh, then "stationary random process" means the PDFs are the same at all times, and that means "identically distributed"(i.d.). But it's not i.i.d. unless the all PDFs at all times are independent of each other. Did I understand correctly? Thank you!! :)👍

(Seems like RU-vid deleted my comments since I put some external links... So I rewrite without the links... c'mmon RU-vid..!) For future ref) Stationary RP has the same PDFs for all times(i.d.), but NOT vice versa(i.d. doesn't mean stationary RP). - For an example of i.d. that's not stationary RP, (example 1) see the comment of the question "Example of identically distributed process, not stationary" in StackExchange/Mathematics. (example 2) see kccu's answer for the question ["strong stationary process" equivalent to a process with "indentically distributed" random variables?] in StackExchange/Mathematics. - For an example of stationary RP that's not i.i.d.(only i.d.), see Joe Blitzstein's answer to this question: "If a stochastic process is strictly stationary, does it imply that the random variables in the process are independent and identically distributed?" in Quora.

Yes. I probably should have mentioned that in the video.

So, a few things here. First of all, what do you mean by "truely random"? It's not a term that gets used by mathematicians, because it's a bit too vague. It sounds like you mean "every possible outcome has equal probability". We call that a "random variable with a uniform probability distribution". We use the term "random" for things where the outcome is unknown ahead of time. It's either "random" or it's not. There are no degrees of "randomness". ie. nothing is "truely random" - it's just "random". Second thing to say is that a "process" does not "generate a random variable". A process is made up of a time-sequence of random variables. Perhaps you are thinking about an actual "realisation" of a process. In other words a measured outcome/signal over time (for example the maximum temperature measurements in Sydney over the past 1000 days). But the problem is, time only goes forward. We can't repeat the "experiment" and try those 1000 days over again, to see whether those particular 1000 days are truely representative of Sydney's long term weather, and test if a different outcome would have happened if we'd had those 1000 days over again. Basically we need to accept that probability distributions are only models of what we believe reality to be. Now we're getting philosophical ... but what else can we do? Random variables are at the interface between "science" and "belief" - that's what makes them so interesting.

This video might help: "What is a Random Variable?" ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-MM6QM3y8pvI.html

0,1 and 2 are not random variables. They are not random, and they are not variables. They are numbers. They don't vary. 0 is always 0. 1 is always 1. 2 is always 2. A "variable" is something that can take on multiple values. So, in your example, you might decide to define the random variable "X" to be the addition of the outcomes of two coin tosses. Then X will have the value 0 if two tails are tossed, it will have the value 1 if one tail and 1 head is tossed, and it will have the value 2 if two heads are tossed. The value of X will vary, depending on the outcome of the experiment. Therefore it is a "variable". And since the outcomes are random, it is a "random variable".

See this video for more explanation: "What is a Random Variable?" ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-MM6QM3y8pvI.html

@@iain_explains thanks

@@iain_explains I want to know about continous random variable

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