Probability Density Function

Probability Density Function#

Definition #

As mentioned in Definition 101, a continuous random variable $X$ is defined by its probability density function (PDF) $f (x)$ or its cumulative distribution function (CDF) $F (x)$ .

Definition 102 (Probability Density Function)

The probability density function (PDF) of a random variable $X$ is a mapping $f_{X} (x)$

\begin{array}{r} \begin{aligned} f_{X} : X (Ω) & \to R \\ X (Ω) ∋ x & \mapsto f_{X} (x) \end{aligned} \end{array}

which satisfies the following properties [Chan, 2021]:

Non-negativity: $f_{X} (x) \geq 0$ for all $x \in Ω$ .
Unity: $\int_{Ω} f_{X} (x) d x = 1$ .
Measure of a set: $P [{x \in A}] = \int_{A} f_{X} (x) d x$ for all $A \subseteq Ω$ .

Remark 40 (Probability Density Function)

The probability density function $f (x)$ of a continuous random variable is similar to the probability mass function $p (x)$ of a discrete random variable, but with two key differences:

Unlike the PMF $p_{X} (x)$ , the PDF $f_{X} (x)$ is not a probability. The PDF $f_{X} (x)$ is a density, which is a measure of the probability of a random variable $X$ taking on a value $x$ . The higher the density, the more likely it is that $X$ takes on the value $x$ (or a value close to $x$ ).
This means that the PDF $f_{X} (x)$ is not necessarily bounded and can be greater than 1.

Notice that the definition of PDF above did not “explicitly” mention the probability of a random variable $X$ , instead it just mentions that the measure of a set to be $P [{x \in A}] = \int_{A} f_{X} (x) d x$ .

The author further mentioned if we are dealing with 1-dimensional data (on the real line), then we can then give a more intuitive definition of the PDF.

Definition 103 (Probability Density Function (1-dimensional))

Let $X$ be a continuous random variable on the real line $R$ . The probability density function (PDF) of $X$ is a mapping $f_{X} (x)$

\begin{array}{r} \begin{aligned} f_{X} : X (Ω) & \to R \\ X (Ω) ∋ x & \mapsto f_{X} (x) \end{aligned} \end{array}

that when when integrated over an interval $[a, b] \subseteq R$ , yields the probability of observing a value of $X$ in the interval $[a, b]$ (i.e $a \leq X \leq b$ ):

P [a \leq X \leq b] = \int_{a}^{b} f_{X} (x) d x

Notice that we have replaced $\int_{A}$ with $\int_{a}^{b}$ where $A = [a, b]$ .

Definition 104 (Zero Measure)

The probability of a continuous random variable $X$ taking on a value $x$ is zero:

P [X = x] = 0

Remark 41 (Open Equals Closed Interval)

By Definition 104, all isolated points have zero measure in the continuous space and therefore the probability of an open interval $(a, b)$ is equivalent to the probability of a closed interval $[a, b]$ .

More concretely, let $P [[a, b]] = P [a \leq X \leq b]$ , then

\begin{array}{r} P [[a, b]] = P [(a, b)] = P [(a, b]] = P [[a, b)] \end{array}

This may not hold when the PDF of $f_{X} (x)$ has a delta function at $a$ or $b$ [Chan, 2021].

References and Further Readings #

Chan, Stanley H. “Chapter 4.1. Probability Density Function.” In Introduction to Probability for Data Science, 172-180. Ann Arbor, Michigan: Michigan Publishing Services, 2021.

Probability Density Function

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Probability Density Function#

Definition#

References and Further Readings#

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