Probability density function

A probability density function (PDF) is a function associated with a continuous Random variable that gives the Probability that the variable, when measured, falls between a certain range of values. The PDF describes the Probability distribution that the variable follows, though some care should be taken when interpreting individual values of the PDF.

Formally, for a random variable $X$, its probability density function $f_{X}(x)$ is a non-negative Lebesgue-integrable function such that the probability of $X$ falling in the range $[a,b]$ is

It must also be normalized, that is, integrate to one over the entire sample space to satisfy the definition of probability:

Unlike the Probability mass function, the PDF does not return probability values: it must be integrated to find the probabilities. In fact, unlike the PMF, the density's image is not limited between 0 and 1 and may contain values greater than 1. As such, the values of a PDF do not qualify as a probability by definition. The integral of the density, however, does have an image of $[0,1]$ and fits the requirements of a probability, hence the normalization requirement. It is therefore convenient to define the integral of the PDF

This integral is known as the cumulative distribution function of $X$. It represents the probability that $X$ will be lower than $x$.

Transformations#

Given a continuous random variable $X$ of PDF $f_{X}$ and an invertible transformation $g(x)$, we can define the transformed random variable as $Y=g(X)$. The PDF of $Y$ is

The same definition applies to a continuous random vector $\mathbf{X}$ that is invertibly transformed into another random vector $\mathbf{Y}=g(\mathbf{X})$, except that the derivative becomes the determinant of the Jacobian of $\mathbf{X}$ with respect to $\mathbf{Y}$: