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Probability density function

A probability density function (pdf) describes how probability is distributed over continuous outcomes. It links to the cumulative distribution, expectations, common examples, and distinctions from discrete mass functions.

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A probability density function (commonly abbreviated pdf) describes how probability is assigned across the possible values of a continuous random variable. Informally, a pdf is a nonnegative function f(x) whose integral over any interval gives the probability that the variable falls in that interval. The total integral of f over its domain equals 1, which enforces that some outcome occurs. Unlike discrete probabilities, a pdf does not give probability at single points; instead probabilities of precise values are zero and only integrals over ranges are meaningful.

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4 continuous probability density functions
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Definition and basic properties

Formally, for a continuous distribution with pdf f(x) the cumulative distribution function (cdf) is defined by F(x) = \int_{-\infty}^{x} f(t)\,dt. For any real numbers a < b, P(a \le X \le b) = \int_{a}^{b} f(x)\,dx. Important properties are: f(x) \ge 0 for all x; \int_{-\infty}^{\infty} f(x)\,dx = 1; the pdf may be zero outside its support and may take values greater than 1 provided the integral remains 1. The pdf is determined only up to changes on sets of measure zero.

Uses and calculations

Dense functions are central to computing expectations and moments: the expected value of X is E[X] = \int x f(x)\,dx and variance is Var(X) = \int (x - E[X])^2 f(x)\,dx. Pdfs appear in likelihood functions for parameter estimation, in Bayesian posterior densities, and in nonparametric methods such as kernel density estimation. Conditional and joint pdfs extend the idea to multivariate settings and form the basis of continuous statistical modelling in statistics.

Examples and common families

  • Uniform: constant density on a finite interval; every subinterval of equal length has equal probability.
  • Normal (Gaussian): bell-shaped, defined on the whole real line; characterized by mean and variance and widely used in practice.
  • Exponential: positive support with a decreasing density, often used for waiting times and lifetimes.

These examples illustrate different shapes and supports a pdf can have. Many other families (gamma, beta, t, etc.) model skewness, heavy tails, or bounded ranges.

Distinctions and notable facts

A pdf differs from a probability mass function (pmf) used for discrete variables: a pmf assigns probability to individual outcomes, while a pdf assigns density per unit length. In some real-world models a distribution is mixed, containing both a pdf part and discrete atoms. Computation of probabilities always involves the integral of the pdf. For further reading on related definitions and rigorous measure-theoretic foundations see introductory texts and reference articles on probability distributions and on the distinction between continuous and discrete models in probability theory (continuous models).

Historically the formal use of densities grew as calculus and probability theory matured in the 18th and 19th centuries. Modern treatments emphasize the role of measure theory in defining densities and expectations. To explore applications and computational methods, consult resources on statistical inference and density estimation techniques (probability, statistics, random variable, function).

Related topics include joint and marginal pdfs for multivariate variables, transformation of variables (change-of-variable formula), and numerical integration methods for evaluating probabilities when closed-form integrals are unavailable.

The probability that a random variable takes a value between aand bcorresponds to the content of the area Sunder the graph of the probability density function f.

Definition

Probability densities can be defined in two ways: once as a function from which a probability distribution can be constructed, the other time as a function derived from a probability distribution. The difference, then, is the direction of approach.

On the construction of probability measures

Given a real function

{\displaystyle f\colon \mathbb {R} \to \mathbb {R} }, for which holds:

  • fis nonnegative, that is, f(x)\geq 0for all x\in \mathbb {R} .
  • fis integrable.
  • fis normalized in the sense that

{\displaystyle \int _{-\infty }^{\infty }f(x)\,\mathrm {d} x=1}.

Then f is called a probability density function and defined by

{\displaystyle P([a,b]):=\int _{a}^{b}f(x)\,\mathrm {d} x}

a probability distribution on the real numbers.

Derived from probability measures

Given a probability distribution Por a real-valued random variable X.

If there exists a real function fsuch that for all {\displaystyle a\in \mathbb {R} }

{\displaystyle P((-\infty ,a])=\int _{-\infty }^{a}f(x)\,\mathrm {d} x}

respectively

{\displaystyle P(X\leq a)=\int _{-\infty }^{a}f(x)\,\mathrm {d} x}

holds, then f is called the probability density function of Por of X.

Examples

A probability distribution that can be defined by a probability density function is the exponential distribution. It has the probability density function

{\displaystyle f_{\lambda }(x)={\begin{cases}\displaystyle \lambda \mathrm {e} ^{-\lambda x}&x\geq 0\\0&x<0\end{cases}}}

Here, λ \lambda >0a real parameter. In particular, the probability density function for parameter λ \lambda >1 at the point x=0 exceeds the function value 1 , as described in the introduction. That {\displaystyle f_{\lambda }(x)}really a probability density function follows from the elementary integration rules for the exponential function, positivity and integrability of the exponential function are clear.

A probability distribution from which a probability density function can be derived is the continuous uniform distribution on the interval [0,1]. It is defined by

{\displaystyle P([a,b])=b-a}for {\displaystyle a\leq b}and {\displaystyle a,b\in [0,1]}

Outside the interval all events get the probability zero. We are now looking for a function ffor which

{\displaystyle \int _{a}^{b}f(x)\,\mathrm {d} x=P([a,b])=b-a}

holds if {\displaystyle a,b\in [0,1]}. The function

{\displaystyle f(x)=1}

satisfies this. It is then continued outside the interval [0,1]by zero to integrate easily over any subset of the real numbers. A probability density function of the continuous uniform distribution would thus be:

{\displaystyle f(x)={\begin{cases}\displaystyle 1&{\text{ falls }}x\in [0,1]\\0&{\text{ sonst }}\end{cases}}}

Similarly, the probability density function would be

{\displaystyle f(x)={\begin{cases}\displaystyle 1&{\text{ falls }}x\in (0,1)\\0&{\text{ sonst }}\end{cases}}}

possible, since both differ only on a Lebesgue null set and both satisfy the requirements. One could create any number of probability density functions just by modifying the value at a point. In fact, this does not change the property of the function being a probability density function, since the integral ignores these small modifications.

For more examples of probability densities, see the list of univariate probability distributions.

Probability density functions of the exponential distribution for different parameters.
Probability density functions of the exponential distribution for different parameters.

Questions and answers

Q: What is a probability density function?

A: A probability density function is a function that characterizes any continuous probability distribution.

Q: How is the probability density function of a random variable X written?

A: The probability density function of X is sometimes written as f_X(x).

Q: What does the integral of the probability density function represent?

A: The integral of the probability density function represents the probability that a given random variable with the given density is contained in an interval provided.

Q: Is the probability density function always non-negative throughout its domain?

A: Yes, by definition, the probability density function is non-negative throughout its domain.

Q: Does integrating over an interval sum up to 1?

A: Yes, integrating over an interval sums up to 1.

Q: What type of distribution does a Probability Density Function characterize?

A: A Probability Density Function characterizes any continuous probability distribution.

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