Probability density function

In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would be equal to that sample.^[2]^[3] Probability density is the probability per unit length, in other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample.

More precisely, the PDF is used to specify the probability of the random variable falling within a particular range of values, as opposed to taking on any one value. This probability is given by the integral of this variable's PDF over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and the area under the entire curve is equal to 1.

The terms probability distribution function and probability function have also sometimes been used to denote the probability density function. However, this use is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over general sets of values or it may refer to the cumulative distribution function, or it may be a probability mass function (PMF) rather than the density. "Density function" itself is also used for the probability mass function, leading to further confusion.^[4] In general though, the PMF is used in the context of discrete random variables (random variables that take values on a countable set), while the PDF is used in the context of continuous random variables.

Absolutely continuous univariate distributions[edit]

A probability density function is most commonly associated with absolutely continuous univariate distributions. A random variable $X$ has density $f_{X}$ , where $f_{X}$ is a non-negative Lebesgue-integrable function, if: $\Pr[a\leq X\leq b]=\int _{a}^{b}f_{X}(x)\,dx.$

Hence, if $F_{X}$ is the cumulative distribution function of $X$ , then: $F_{X}(x)=\int _{-\infty }^{x}f_{X}(u)\,du,$ and (if $f_{X}$ is continuous at $x$ ) $f_{X}(x)={\frac {d}{dx}}F_{X}(x).$

Intuitively, one can think of $f_{X}(x)\,dx$ as being the probability of $X$ falling within the infinitesimal interval $[x,x+dx]$ .

Further details[edit]

Unlike a probability, a probability density function can take on values greater than one; for example, the continuous uniform distribution on the interval $[0, 1/2]$ has probability density $f (x) = 2$ for $0 \leq x \leq 1/2$ and $f (x) = 0$ elsewhere.

The standard normal distribution has probability density $f(x)={\frac {1}{\sqrt {2\pi }}}\,e^{-x^{2}/2}.$

If a random variable $X$ is given and its distribution admits a probability density function $f$ , then the expected value of $X$ (if the expected value exists) can be calculated as $\operatorname {E} [X]=\int _{-\infty }^{\infty }x\,f(x)\,dx.$

Not every probability distribution has a density function: the distributions of discrete random variables do not; nor does the Cantor distribution, even though it has no discrete component, i.e., does not assign positive probability to any individual point.

A distribution has a density function if and only if its cumulative distribution function $F (x)$ is absolutely continuous. In this case: $F$ is almost everywhere differentiable, and its derivative can be used as probability density: ${\frac {d}{dx}}F(x)=f(x).$

If a probability distribution admits a density, then the probability of every one-point set ${a}$ is zero; the same holds for finite and countable sets.

Two probability densities $f$ and $g$ represent the same probability distribution precisely if they differ only on a set of Lebesgue measure zero.

In the field of statistical physics, a non-formal reformulation of the relation above between the derivative of the cumulative distribution function and the probability density function is generally used as the definition of the probability density function. This alternate definition is the following:

If $dt$ is an infinitely small number, the probability that $X$ is included within the interval $(t, t + dt)$ is equal to $f (t) dt$ , or: $\Pr(t<X<t+dt)=f(t)\,dt.$

Link between discrete and continuous distributions[edit]

It is possible to represent certain discrete random variables as well as random variables involving both a continuous and a discrete part with a generalized probability density function using the Dirac delta function. (This is not possible with a probability density function in the sense defined above, it may be done with a distribution.) For example, consider a binary discrete random variable having the Rademacher distribution—that is, taking −1 or 1 for values, with probability 1⁄2 each. The density of probability associated with this variable is: $f(t)={\frac {1}{2}}(\delta (t+1)+\delta (t-1)).$

More generally, if a discrete variable can take $n$ different values among real numbers, then the associated probability density function is: $f(t)=\sum _{i=1}^{n}p_{i}\,\delta (t-x_{i}),$ where $x_{1},\ldots ,x_{n}$ are the discrete values accessible to the variable and $p_{1},\ldots ,p_{n}$ are the probabilities associated with these values.

This substantially unifies the treatment of discrete and continuous probability distributions. The above expression allows for determining statistical characteristics of such a discrete variable (such as the mean, variance, and kurtosis), starting from the formulas given for a continuous distribution of the probability.

Families of densities[edit]

It is common for probability density functions (and probability mass functions) to be parametrized—that is, to be characterized by unspecified parameters. For example, the normal distribution is parametrized in terms of the mean and the variance, denoted by $\mu$ and $\sigma ^{2}$ respectively, giving the family of densities $f(x;\mu ,\sigma ^{2})={\frac {1}{\sigma {\sqrt {2\pi }}}}e^{-{\frac {1}{2}}\left({\frac {x-\mu }{\sigma }}\right)^{2}}.$ Different values of the parameters describe different distributions of different random variables on the same sample space (the same set of all possible values of the variable); this sample space is the domain of the family of random variables that this family of distributions describes. A given set of parameters describes a single distribution within the family sharing the functional form of the density. From the perspective of a given distribution, the parameters are constants, and terms in a density function that contain only parameters, but not variables, are part of the normalization factor of a distribution (the multiplicative factor that ensures that the area under the density—the probability of something in the domain occurring— equals 1). This normalization factor is outside the kernel of the distribution.

Since the parameters are constants, reparametrizing a density in terms of different parameters to give a characterization of a different random variable in the family, means simply substituting the new parameter values into the formula in place of the old ones.

– Estimate of an unobservable underlying probability density function

Density estimation

– Estimator

Kernel density estimation

– Function related to statistics and probability theory

Likelihood function

List of probability distributions

– Complex number whose squared absolute value is a probability

Probability amplitude

– Discrete-variable probability distribution

Probability mass function

Secondary measure

Atomic orbital

(1979). Probability and Measure. New York, Toronto, London: John Wiley and Sons. ISBN 0-471-00710-2.

Billingsley, Patrick

; Berger, Roger L. (2002). Statistical Inference (Second ed.). Thomson Learning. pp. 34–37. ISBN 0-534-24312-6.

Casella, George

Stirzaker, David (2003). . Cambridge University Press. ISBN 0-521-42028-8. Chapters 7 to 9 are about continuous variables.