Clarification on meaning of "Gaussian random variable"

Question

When my lecturer uses the word Gaussian random variable, he always writes the pdf of the Gaussian instead of the random variable itself.

For example,

given a random variable $X$ Gaussian, $f_X(x) = \frac{1}{\sqrt{2\pi \sigma^2}}\exp(-\frac{1}{2}\sigma^{-2} x^2)$

I am so confused as to why can't we talk about $X$ itself without using pdf?

Recall the definition of random variable is a function $X$ that maps outcomes to the real numbers i.e. $X: \zeta \to \mathbb{R}$

In this case, given $X$ is Gaussian, what are the outcomes of the sample space and what is the value that $X$ has assigned to outcomes of the sample space?

Answer 1

The domain of $X$ is $\Omega$, which can literally be anything. For example $X(\text{ I roll a six}) = 1$

So when I roll a six, the random value $X$ takes on the value $1$. So random variables "transfer" events into real numbers (in reality, things are more general; you need not to have $\mathbb R$ )

Now the point is that you have no idea "how often " does $X$ takes on the value $1$, because it depends on what happens in the real world ie. If I roll a six or not.

So $X$ is actually deterministic, it associates to certain events certain values. The idea then is that if you know the probabilities of certain events happening, you know the probability that $X$ takes on a certain value.

So technically we defin $\Omega$, a probability $P$ on $\Omega$, which induces through the relation $X$ a probability $P_X$ on the reals. In practice we just assign $P_X$ (for example a gaussian ) and we don't specify $\Omega$ and $P$

Answer 2

There is infinitely many random variables with the same pdf (Gaussian in this case). In probability theory is very common to study all the random variables with the same pdf, so there is no need to talk about a specific random variable. What really matters is to understand the general properties of these random variables, and one way to study these properties is studying their pdf.

Answer 3

For each expected value and each variance there is only one normal distribution, so if one says that $X$ is normally distributed with expected value $\mu$ and variance $\sigma^2$ then one has said what its distribution is. Often one writes something like this:

Suppose $X\sim N(\mu,\sigma^2)$. Then [conclusions follow].

That does not involve mentioning a density, and that is what is usually done.

The outcomes in the sample space can be any of a number of different things. Sometimes one considers an infinite sequence $X_1,X_2,X_3,\ldots$ of independent random variables, and an outcome is an infinite sequence of numbers. In such a case normally the value of $X_1$ at an outcome that is a particular sequence would be the first number in the sequence and $X_2$ would be the second one, and so on.