An RV $X:\Omega\to \R$ defines a new probability function $\P'$ on a new sample space $\Omega'=\mathbb{R}$: \begin{align} \tag{*} \P'(E) = \P(X\in E), \qquad E\subset \mathbb{R}=\Omega'. \end{align} Verifying that $\P'$ satisfies the three probability axioms is straightforward. We can often leverage the conceptually simpler $(\R,\P')$ in computing probabilities $\P(A)$ of events $A=\{X\in E\}$, ignoring $(\Omega,\P)$.

An RV can be discrete, continuous, or neither. If $\{X(\omega):\omega\in\Omega\}$ is finite or countably infinite, $X$ is discrete. This happens in particular if $\Omega$ is finite or countably infinite.

The cdf is useful for both discrete and continuous RVs, the pmf is useful for discrete RVs only, and the pdf is useful for continuous RVs only. Notice that in each of the three functions above, we denote the applicable RV with an uppercase subscript and the function argument with a lowercase letter, and the two usually correspond (for example, $p_X(x)=\P(X=x)$).

We have the following relationship between the cdf and the pdf \begin{align*} f_X(r)&= \begin{cases} F_X'(r)& F_X'(r)\text{ exists}\\ 0 & \text{otherwise}\end{cases} \\ F_X(x)&=\int_{-\infty}^x f_X(r)\,dr. \end{align*} where the first equation follows from the definition of a pdf and the second line follows from the fundamental theorem of calculus (see Chapter E).

D = data.frame(x = c(-1, 0, 1, 2, 3, 4), y = c(0, 1/8,
    3/8, 3/8, 1/8, 0))
qplot(x, y, data = D, xlab = "$x$", ylab = "$p_X(x)$") +
    geom_linerange(aes(x = x, ymin = 0, ymax = y))

par(cex.main = 1.5, cex.axis = 1.2, cex.lab = 2)
plot(ecdf(c(0, 1, 1, 1, 2, 2, 2, 3)), verticals = FALSE,
    lwd = 3, main = "$F_X(x)$ as a function of $x$",
    xlab = "", ylab = "")
grid()

In the previous figure, the filled circles above clarify the value of $F_X$ at the discontinuity points above, for example $F_X(3)=1$. Note how the $F_X$ above is monotonic increasing from 0 to 1, and is continuous from the right, but not continuous from the left.

x = seq(-0.5, 1.5, length = 100)
y = x^2
y[x < 0] = 0
y[x > 1] = 1
z = 2 * x
z[x < 0] = 0
z[x > 1] = 0
D = stack(list(`$f_R(r)$` = z, `$F_R(r)$` = y))
names(D) = c("probability", "Function")
D$x = x
qplot(x, y = probability, geom = "line", xlab = "$r$",
    ylab = "", lty = Function, color = Function, data = D,
    size = I(1.5))

The reason for the infimum in the definition above is that $F_X$ may not be a one-to-one function, and there may be multiple values $x$ for which $F_X(x)=r$. Recall that for a continuous $X$, the cdf $F_X$ is continuous and monotonic increasing from 0 at $-\infty$ to 1 at $\infty$ (see Definition B.2.5). If the cdf is strictly monotonically increasing (see Definition B.2.5), $F_X$ is one-to-one and hence invertible and $Q_X(r)=F_X^{-1}(r)$. The value, $Q_X(1/2)$, at which 50% of the probability mass lies above and 50% of the probability mass lie below is called the median.

Since the pdf is the derivative of the cdf we have \[f_X(x)=\lim_{\Delta\to 0}\frac{F_X(x+\Delta)-F_X(x)}{\Delta}.\] This implies that for a small positive $\Delta$ \begin{align*} f_X(x) &\approx \frac{\P(x < X < x+\Delta)}{\Delta}, \quad\text{or}\\ f_X(x) \Delta &\approx \P(x < X < x+\Delta). \end{align*} The above derivations shows that $f_X(x)\Delta$ is approximately the probability that $X$ lies within an interval of width $\Delta$ around $x$.