## Probability Mass Function (PMF)
The PMF is suited for *discrete random variables* $X$. It assigns every realization of $X$ a **probability** in the interval $[0,1]$. The name *mass* refers to how much it influences the corresponding outcome of $X$:
$
p_X(x)=P(X=x).
$
In addition to each probability mass being in the interval $[0,1]$, the PMF must “integrate” to 1:
$
\sum_x p_X(x) = 1.
$
## Probability Density Function (PDF)
The PDF is suited for *continuous random variables* $X$. It is a **density function**, as opposed to a **mass function**, because it represents **where** the mass is, not **what** the mass is.
In fact, the probability of all random variables at any point $x$ is **always** $0$! The reason is simple: there is an infinite amount of real-valued numbers in any interval, and the sum of all their probabilities must add up to 1.
>[!important]
> *Even for fixed intervals*, a random variable has an infinite amount of possible values, whose probabilities must sum up to 1.
To compute the probability of a certain realization of $X$, we integrate the PDF over an interval :
$
\begin{align}
p(X=x) &= \int_{-\infty}^\infty xp(x)dx, \\
p(a < X <b) &= \int_a^b xp(x)dx.
\end{align}
$
In order for the PDF to be valid, it must integrate to 1:
$
\int_{-\infty}^\infty xp(x)dx = 1.
$