what's the difference between variable and process from a statistical point of view?

Question

I'm reading a tutorial stochastic process: ergodicity and temporal averages and I'm totally confused. It is said that:

Suppose an IID random process whose marginal PDF is Gaussian with mean and variance equal 1. let $X_i[18]$ denote the $i$th realization at time $n = 18$, then by definition $$\lim_{M\to +\infty}\frac{1}{M}\sum_{m=1}^M x_m[18]=E[X[18]]=\mu_X[18]=\mu=1$$ is called averaging down the ensemble and consequently is just a restatement of our usual notion of the expected value of a random variable
However, if we are given only a single realization such as $x_i[n]$, then it seems reasonable that $$\hat\mu_N=\frac{1}{N}\sum_{n=0}^{N-1}x_1[n]$$ should also converge to $\mu$ as $N\to\infty$

Since I am not educated enough in stochastic processes, I'm stuck with lots of maybe elementary problems that make me confused and I don't know where to start on the web to study about because whenever I google the term, I encounter with some new terms that I don't understand the meaning of them so help me as if you are illustrating these concepts to a layman or your grand mum as Einstein says.
My knowledge in statistics is in the level of what I have studied in the first year of college for the course probability and statistics in engineering and few other courses for the adjustment of errors. I've never even heard of terms such ergodicity and etc. But I do need the concept in my study now.

My questions:

1- what does Independent and identically distributed random process mean?  

2-what's the difference of random process with random variable?

3-Is there a difference between stochastic process and random process? 

4-Is there a difference between stochastic variable and random variable?  

5-what does the realization of a random process mean?  

6-Is the first equation equal to `ensemble average`? 

please provide examples and simplification in your answers as much as you can as if you are illustrating the concept to a layman

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For example:
1-Tossing the die is an example of a random process;
2-The number on top of the die is an example of random variable.
Could you please explain me how to compute the ensemble average and temporal average of tossing a die? What does realization mean in this specific example?

Answer 1

First, random and stochastic are synonyms.

A stochastic process is a collection of random variables $X=(X_i)_{i\in I}$.

You have two important cases:

• the discrete case: $I=\mathbb{N}$ or $\mathbb{Z}$ (or a subset of those). Then the process is a sequence of random variables.
• the continuous case : $I=\mathbb{R}$ or $[0,+\infty )$ for example. Then the stochastic process is a random function.

As you may know, a random variable is a measurable function $\Omega\rightarrow S$ where $S$ is a measurable space. Then there are two points of view regarding stochastic processes. You can see them as a collection of random variables indexed by the set $I$. Or you can see them as a single random variable with a bigger $S$.

For example, let's say you have a stochastic process $X=(X_n)_{n\in\mathbb{N}}$ with $X_n\in\mathbb{R}$. Then you can see $X$ as a random variable with $S$ the set of all real-valued sequences. You can write $X=X(\omega ,n)$. If you fix $\omega$, you obtain a real-valued sequence which is called a realization of $X$ (or a trajectory). If you fix $n$ (the time for example), you obtain the real-values random variable $X_n$.