Probability: mathematically what does it mean to say "let $X$ be a random variable WITH a cdf/pdf"

Question

I don't quite understand what people mean by let "$X$ be a random variable WITH a cdf/pdf". For example, there is a question that says: "Let X be a random variable with the 3-parameter Weibull pdf and cdf"

Suppose I say: Let $X$ be a random variable with Gaussian CDF.

What does that mean exactly? $X$ is a function that maps from the event space to a real number. What does it mean to "connect" it, "link" it, "equip" it, WITH a CDF? It is already a function, what does it mean to let it hook up with another function, why don't we just deal with $X$ directly.

Then I look at the CDF:

$f_X(x) = \int_A \exp(-x^2/2)dx$

I ask myself: Is "$\exp(-x^2/2)$" part the random variable? No. Is "$\int_A \exp(-x^2/2)dx$" the random variable? No. Is little $x$ is the random variable? No. What is the difference if I wrote $f(x)$ instead of $f_X(x)$? Nothing happens.

What does the CDF/PDF have to do with the random variable exactly? How do you know which CDF/PDF a random variable "has"?

Answer 1

Yes $X$ is a function from the event space $\Omega$ to the reals $\mathbb{R}$, but there is a bit more: $\Omega$ is a probability space, that is, it has a $\sigma$-algebra and a probability measure. If these terms are unfamiliar to you, loosely speaking this is just saying that every event has a probability associated with it [that satisfies some reasonable properties, e.g., probability of union of two disjoint events is the sum of their respective probabilities].

Note that this induces a probability measure on $\mathbb{R}$ as well. Given any interval $(-\infty,t]$ (or more generally any open set), we can define its probability as the probability of the preimage $X^{-1}((-\infty,t])$, that is, the probability of the subset of $\Omega$ that $X$ maps to $(-\infty,t]$. We often denote this simply as $\mathbb{P}(X \le t)$.

Without this notion of probability, all you would have is a map; you wouldn't have any notion of questions like "what is the probability that $X$ lies in the interval $[0,3]$?" More concretely, if $X$ is the outcome of a roll of a die, then all you have is a map to $\{1,\ldots,6\}$ and no notion of how likely each outcome is. The randomness comes from the underlying probability measure.

Now, if one says "$X$ has CDF $F$," then this is just specifying the underlying probability measure, i.e. it tells you $F(t) := \mathbb{P}(X \le t)$ for any $t$. It turns out that if we know this information, we can find the probability of any Borel set.

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Regarding your last few questions ("What is the CDF/PDF of a random variable?") here are the definitions explicitly:

The CDF of a random variable $X$ is a function $F:\mathbb{R} \to [0,1]$ defined by $F(t) := \mathbb{P}(X \le t)$.

Not every random variable has a PDF, but if its CDF is differentiable, then its PDF is defined to be $f(x) := \frac{d}{dt}F(t)$. By calculus, the PDF satisfies $F(t)=\int_{-\infty}^t f(x) \mathop{dx}$.

Answer 2

A random variable formally is a measurable map from a probability space $(\Omega, \sigma, P)$ to $(\Bbb{R}, B(\Bbb{R}), \mu)$ where $\mu$ is Lebesgue measure and $B(\Bbb{R})$ is the borel $\sigma$-algebra.

If $P \ll \mu$ (read $P$ is absolutely continuous wrt $\mu$. Meaning $\mu(S)=0\implies P(S)=0$) and $\Omega$ is $\sigma$-finite, then the Radon Nikodym derivative, $f$ exists and is called the probability density function.

If you want a little more elementary explanation. By "measurable map" we mean that $P(X\leq x)$ is always defined. We simply define the new function $F(x)=P(X\leq x)$ and if the derivative of $F$ exists, then $f$ is the PDF.