Probability Density Function (PDF)

For a huge class of probability distributions or probability measures, the probability of an event (i.e., the random variable falling into a certain set $A$ ) is found by integrating a specific function—the probability density function (PDF), denoted $f (x)$ —over that set. The "integration" itself is done with respect to the Lebesgue measure ( $λ$ ).

A probability measure $P$ is absolutely continuous with respect to the Lebesgue measure $λ$ if it can be represented as:

P (A) = \int_{A} f (x) d λ (x)

This is usually written in the more familiar elementary calculus notation:

P (A) = \int_{A} f (x) d x

For $f (x)$ to be a valid PDF, it must satisfy two conditions:

Non-negativity: $f (x) \geq 0$ for all $x$ .
Normalization: The total integral over the entire space must be 1.

\int_{R} f (x) d x = 1

Think of the Lebesgue measure $d x$ as providing the fundamental notion of "length" on the real line. The PDF $f (x)$ then acts as a weighting factor, stretching or shrinking that length to assign a probability.

Famous Examples

1. Normal (Gaussian) Distribution

The famous "bell curve" is the PDF for the Normal distribution. To find the probability that a normally distributed random variable $X$ falls between $a$ and $b$ , you integrate its PDF:

PDF: $f (x) = \frac{1}{σ \sqrt{2 π}} e^{- \frac{(x - μ)^{2}}{2 σ^{2}}}$
Probability: $P (a \leq X \leq b) = \int_{a}^{b} \frac{1}{σ \sqrt{2 π}} e^{- \frac{(x - μ)^{2}}{2 σ^{2}}} d x$

2. Exponential Distribution

Used to model waiting times, its probability is also the integral of its PDF.

PDF: $f (x) = λ e^{- λ x}$ for $x \geq 0$ (and $0$ otherwise).
Probability: $P (X > t) = \int_{t}^{\infty} λ e^{- λ x} d x = e^{- λ t}$

3. Uniform Distribution on $[a, b]$

This is the simplest case. The PDF is a constant over the interval, meaning the probability is directly proportional to the length of the sub-interval (its Lebesgue measure).

PDF: $f (x) = \frac{1}{b - a}$ for $x \in [a, b]$ (and $0$ otherwise).
Probability: $P (c \leq X \leq d) = \int_{c}^{d} \frac{1}{b - a} d x = \frac{d - c}{b - a}$ (for $a \leq c \leq d \leq b$ ).

It's crucial to know that not all distributions work this way.

Discrete Distributions

Discrete distributions, like the Poisson or Binomial, are singular with respect to the Lebesgue measure.

Their probability mass is concentrated on a countable set of points (e.g., the integers ${0, 1, 2, . . .}$ ). This set has a Lebesgue measure of zero. So, you cannot find a PDF $f (x)$ to integrate.

Instead, they are defined by a probability mass function (PMF), which gives the probability at each specific point. In the language of measure theory, a discrete distribution is a weighted sum of Dirac measures.