Hi everybody,(adsbygoogle = window.adsbygoogle || []).push({});

I try to figure out connections and differences between random variables (RV), random processes (RP), and sample spaces and have confusions on some ideas you may want to help me.

All sources I searched says that RP assigns each element of a sample space to a time function. I want to give two examples:

a. Roll a six-sided die and observe the number coming out. Sample space is Ω={1,2,3,4,5,6}. Assign a time function x_{i}(t) from the collection {x_{1}(t), x_{2}(t), x_{3}(t), x_{4}(t), x_{5}(t), x_{6}(t)} to each ω_{i}∈Ω and start to roll the die to infinity. At each trial of roll, an outcome from Ω comes out, this also determines an x_{i}(t), and so depending on time, it generates a number. Now, is this a RP? If it is, what is the outcome of the RP?

b. My second example is on rolling a die again. This time one shot experiment is rolling a die infinitely. Eventhough sample space of one shot experiment is {1,2,3,4,5,6}, the sample space Ω subject to RP is, in fact, infinite strings where each digit of which has a number from one to six such as, (1645623...456321234...) is one possible outcome. In this case sample space Ω has infinite number of outcomes. Now I assign again each outcome to a time function x_{i}(t). The time function takes its domain t values such as t=1 first trial, t=2 second trial, t=i i^{th}trial, etc and range x_{i}(t) values as outcomes of the trial at time t. Assume I assigned my first trial outcome (1645623...456321234...) to x_{1}(t) then x_{1}(1)=1, x_{1}(2)=6, x_{1}(3)=4, etc. So, is this a RP? If it is, what is the outcome of this process?

Thank you for taking your time and help.

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# Random Process vs Random Variable vs Sample Space

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