Probability density function

In probability theory, a probability density function, density function, or density of an absolutely continuous random variable, is a function whose value at any given sample in the sample space can be interpreted as providing a relative likelihood that the value of the random variable would be equal to that sample. 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 zero, given there is an infinite set of possible values to begin with. Therefore, 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 a continuous variable's PDF over that range, where the integral is the nonnegative area under the density function between the lowest and greatest values of the range. The PDF is nonnegative everywhere, and the area under the entire curve is equal to one, such that the probability of the random variable falling within the set of possible values is 100%.
The terms probability distribution function and probability function can also 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 rather than the density. Density function itself is also used for the probability mass function, leading to further confusion. In general the PMF is used in the context of discrete random variables, while the PDF is used in the context of continuous random variables. Both PMF and PDF are fundamental concepts in statistical inference.

Example

Suppose bacteria of a certain species typically live 20 to 30 hours. The probability that a bacterium lives 5 hours is equal to zero. A lot of bacteria live for approximately 5 hours, but there is no chance that any given bacterium dies at exactly 5.00... hours. However, the probability that the bacterium dies between 5 hours and 5.01 hours is quantifiable. Suppose the answer is 0.02. Then, the probability that the bacterium dies between 5 hours and 5.001 hours should be about 0.002, since this time interval is one-tenth as long as the previous. The probability that the bacterium dies between 5 hours and 5.0001 hours should be about 0.0002, and so on.
In this example, the ratio / is approximately constant, and equal to 2 per hour. For example, there is 0.02 probability of dying in the 0.01-hour interval between 5 and 5.01 hours, and = 2 hour⁻¹. This quantity 2 hour⁻¹ is called the probability density for dying at around 5 hours. Therefore, the probability that the bacterium dies at 5 hours can be written as dt. This is the probability that the bacterium dies within an infinitesimal window of time around 5 hours, where dt is the duration of this window. For example, the probability that it lives longer than 5 hours, but shorter than, is × ≈ .
There is a probability density function f with f = 2 hour⁻¹. The integral of f over any window of time is the probability that the bacterium dies in that window.

Absolutely continuous univariate distributions

A probability density function is most commonly associated with absolutely continuous univariate distributions. A random variable has density, where is a non-negative Lebesgue-integrable function, if:
Hence, if is the cumulative distribution function of, then:
and
Intuitively, one can think of as being the probability of falling within the infinitesimal interval.

Formal definition

A random variable with values in a measurable space has as probability distribution the pushforward measure X_∗P on : the density of with respect to a reference measure on is the Radon–Nikodym derivative:
That is, f is any measurable function with the property that:
for any measurable set

Discussion

In the [|continuous univariate case above], the reference measure is the Lebesgue measure. The probability mass function of a discrete random variable is the density with respect to the counting measure over the sample space.
It is not possible to define a density with reference to an arbitrary measure. Furthermore, when it does exist, the density is almost unique, meaning that any two such densities coincide almost everywhere.

Further details

Unlike a probability, a probability density function can take on values greater than one; for example, the continuous uniform distribution on the interval has probability density for and elsewhere.
The standard normal distribution has probability density
If a random variable is given and its distribution admits a probability density function, then the expected value of can be calculated as
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 its cumulative distribution function is absolutely continuous. In this case: is almost everywhere differentiable, and its derivative can be used as probability density:
If a probability distribution admits a density, then the probability of every one-point set is zero; the same holds for finite and countable sets.
Two probability densities and 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 is an infinitely small number, the probability that is included within the interval is equal to, or:

Link between discrete and continuous distributions

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. For example, consider a binary discrete random variable having the Rademacher distribution—that is, taking −1 or 1 for values, with probability each. The density of probability associated with this variable is:
More generally, if a discrete variable can take different values among real numbers, then the associated probability density function is:
where are the discrete values accessible to the variable and 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, starting from the formulas given for a continuous distribution of the probability.

Families of densities

It is common for probability density 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 and respectively, giving the family of densities
Different values of the parameters describe different distributions of different random variables on the same sample space ; 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. 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.

Densities associated with multiple variables

For continuous random variables, it is also possible to define a probability density function associated to the set as a whole, often called joint probability density function. This density function is defined as a function of the variables, such that, for any domain in the -dimensional space of the values of the variables, the probability that a realisation of the set variables falls inside the domain is
If is the cumulative distribution function of the vector, then the joint probability density function can be computed as a partial derivative

Marginal densities

For, let be the probability density function associated with variable alone. This is called the marginal density function, and can be deduced from the probability density associated with the random variables by integrating over all values of the other variables:

Independence

Continuous random variables admitting a joint density are all independent from each other if

Corollary

If the joint probability density function of a vector of random variables can be factored into a product of functions of one variable
then the variables in the set are all independent from each other, and the marginal probability density function of each of them is given by