![The event that the arrow ends up pointing at the red section is imagined as the interval $[0.5, 0.75]$.](attachment:ff9c8a00-ac4e-4688-971b-c56ee0e725fb:6b1b789c-3623-46da-9e40-633fb1555faf.png)
To reduce clutter, the PDF of a random variable is denoted with a lower case letter and its CDF with an upper case letter.
Probability over continuous sample spaces. In continuous probability, an event is represented by a continuous interval $\mathcal{I} \sube \Omega$, with size given by the length between its endpoints.
The event that the arrow ends up pointing at the red section is imagined as the interval $[0.5, 0.75]$.
There is no clear way to define a probability mass function on a continuous sample space, since probabilities over any subset $\Omega$ of $\R$ must somehow sum to $1$. The only way for this to occur is if each $\omega \in \Omega$ has an infinitesimal probability, which means any outcome has essentially $0$ probability.
The natural solution, then, is to instead define the probability of outcomes as intervals within the sample space, where outcomes $[w_i, \omega_i + d\omega]$ can be approximated with $f(w_i)dw$, for some probability density function $f$.
Directly analogous to the notion of a discrete random variable $X$, except that its range is given by a continuous subset of $\R$.
Additionally, the notion of independence between two random variables $X$ and $Y$ is exactly the same and can be determined if their joint PDF $f_{XY}(x, y)$ can be decomposed into $f_X(x) f_Y(y)$.
Continuous probability distributions are represented by probability density functions (PDF) and cumulative distribution functions (CDF).
$$ \textbf{Probability Distribution Function} \\ \text{For a real-valued r.v. $X$, its PDF is given as the function $f$ such that:} \\ \forall x\in \R, f(x) \geq 0 \text{ with } \int {-\infty}^{\infty}f(x)dx = 1 \\ \Pr[a \leq X \leq b] := \int{a} ^b f(x)dx $$
$$ \textbf{Cumulative Distribution Function} \\ \text{For a real-valued r.v. $X$, its CDF is given as the function $F$ such that:} \\ f=\frac{dF}{dx} \iff F(x) = \int_{-\infty} ^x f(x)dx \\ \Pr[X \leq x] := F(x) $$
For a probability space where each outcome over some support $[a, b]$ is equally likely, we define the PDF and CDF of a random variable $X$ on that space as follows,