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At a simpler level to my previous question, I wanted to confirm my understanding on Random Process based on Random Variables using an example.

So, I took this example: If we consider a dice, which has six sides denoted by fi (i can be 1,2,3,4,5,6). So, assigning outcomes f1=10,f2=20,f3=30,f4=40,f5=50,f6=60 thereby constructs a Random Variable X.

Now, if we assign the outcomes: f1=15,f2=25,f3=35,f4=47,f5=80,f6=90 at a different time instant t2 and at time instant t3, we assign the outcomes: f1=43,f2=32,f3=16,f4=89,f5=99,f6=56.

So, if I plot this variation on a Graph, I am getting something like this below:

enter image description here

Now does this set of sample functions (Ensemble) constitute a Random Process?

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  • $\begingroup$ What does it mean f1=10,f2=20,f3=30,f4=40,f5=50,f6=60? What is f1? What is your "outcome"? Normally outcome is the result of a trial of a probability experiment. $\endgroup$
    – AlexTP
    May 28, 2017 at 9:06
  • $\begingroup$ @AlexTP - I mean f1,f2,...f6 are the six faces of dice and outcome (result of a trial) could be anyone of those faces. Assigning f1,f2.... f6 a corresponding value forms a random variable (I have just assigned (10*i). I wanted to confirm that if this assignment of values changes over time , does this form a Random Process? $\endgroup$
    – sundar
    May 28, 2017 at 15:33

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Let me explain it in another way. Consider you have 6 different function of time. You only throw your dice once and regarding the outcome you choose on of six functions and the chose one is one outcome of your random process.

Now expand this concept to all possible functions of time instead of only 6 functions and instead of dice throwing consider another form of random event which could append a probability to all (infinite) possible functions. The set of all these possible functions is your ensemble.

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  • $\begingroup$ Thanks. So I see that, in Random Variable we assign a specific value to an outcome. But if we assign a specific value to an outcome (for example if we assign six faces of dice these six functions: (f1=sint,f2=sin2t,f3=sin3t,f4=sin4t,f5=sin5t,f6=sin6t), then the result of an experiment is one outcome (a function of time) from above. Now expanding this to multiple outcomes for an event and multiple functions of time, depending on the outcome, we choose a function of time. The concept remains clear till this stage but I could not understand what "appending a probability" refers to. $\endgroup$
    – sundar
    May 28, 2017 at 15:37
  • $\begingroup$ in dice throwing you have a simple sample space which is finite and countable. but for all possible function its more complicated and number of possible function are too much more! by appending probability I was trying to say each one of these functions has a probability. $\endgroup$
    – Mohammad M
    May 28, 2017 at 17:07
  • $\begingroup$ Thanks..Now i get it.. So, yeah - checking the probability of those functions would be viewed as"Joint pdf". And seeing the probability at a specific time instant on all the outcomes would be a "probability density function" for a single random variable. $\endgroup$
    – sundar
    May 28, 2017 at 17:17

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