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In
probability theory Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...
, 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 In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually den ...
(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. 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 A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
falling ''within a particular range of values'', as opposed to taking on any one value. This probability is given by the
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
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 In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
is defined as a function over general sets of values or it may refer to the
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
, or it may be a
probability mass function In probability and statistics, a probability mass function (sometimes called ''probability function'' or ''frequency function'') is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes i ...
(PMF) rather than the density. "Density function" itself is also used for the probability mass function, leading to further confusion. 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.


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 (i.e., 2%). 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 (probability of living during an interval) / (duration of the interval) is approximately constant, and equal to 2 per hour (or 2 hour−1). For example, there is 0.02 probability of dying in the 0.01-hour interval between 5 and 5.01 hours, and (0.02 probability / 0.01 hours) = 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 (2 hour−1) ''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 (5 hours + 1 nanosecond), is (2 hour−1)×(1 nanosecond) ≈ (using the
unit conversion Conversion of units is the conversion of the unit of measurement in which a quantity is expressed, typically through a multiplicative conversion factor that changes the unit without changing the quantity. This is also often loosely taken to incl ...
nanoseconds = 1 hour). There is a probability density function ''f'' with ''f''(5 hours) = 2 hour−1. The
integral In mathematics, an integral is the continuous analog of a Summation, sum, which is used to calculate area, areas, volume, volumes, and their generalizations. Integration, the process of computing an integral, is one of the two fundamental oper ...
of ''f'' over any window of time (not only infinitesimal windows but also large windows) 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 A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
X has density f_X, where f_X is a non-negative Lebesgue-integrable function, if: \Pr \le X \le b= \int_a^b f_X(x) \, dx . Hence, if F_X is the
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
of X, then: F_X(x) = \int_^x f_X(u) \, du , and (if f_X is continuous at x) f_X(x) = \frac F_X(x) . Intuitively, one can think of f_X(x) \, dx as being the probability of X falling within the infinitesimal interval ,x+dx/math>.


Formal definition

(''This definition may be extended to any probability distribution using the measure-theoretic definition of probability.'') A
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
X with values in a measurable space (\mathcal, \mathcal) (usually \mathbb^n with the
Borel set In mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets ...
s as measurable subsets) has as
probability distribution In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
the pushforward measure ''X''''P'' on (\mathcal, \mathcal): the density of X with respect to a reference measure \mu on (\mathcal, \mathcal) is the Radon–Nikodym derivative: f = \frac . That is, ''f'' is any measurable function with the property that: \Pr \in A = \int_ \, dP = \int_A f \, d\mu for any measurable set A \in \mathcal.


Discussion

In the continuous univariate case above, the reference measure is the
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it c ...
. The
probability mass function In probability and statistics, a probability mass function (sometimes called ''probability function'' or ''frequency function'') is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes i ...
of a
discrete random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' in its mathematical definition refers ...
is the density with respect to the counting measure over the sample space (usually the set of
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
s, or some subset thereof). It is not possible to define a density with reference to an arbitrary measure (e.g. one can not choose the counting measure as a reference for a continuous random variable). Furthermore, when it does exist, the density is almost unique, meaning that any two such densities coincide
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
.


Further details

Unlike a probability, a probability density function can take on values greater than one; for example, the
continuous uniform distribution In probability theory and statistics, the continuous uniform distributions or rectangular distributions are a family of symmetric probability distributions. Such a distribution describes an experiment where there is an arbitrary outcome that li ...
on the interval has probability density for and elsewhere. The standard normal distribution has probability density f(x) = \frac\, e^. If a random variable is given and its distribution admits a probability density function , then the
expected value In probability theory, the expected value (also called expectation, expectancy, expectation operator, mathematical expectation, mean, expectation value, or first Moment (mathematics), moment) is a generalization of the weighted average. Informa ...
of (if the expected value exists) can be calculated as \operatorname = \int_^\infty x\,f(x)\,dx. Not every probability distribution has a density function: the distributions of
discrete random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' in its mathematical definition refers ...
s 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 In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
is absolutely continuous. In this case: is
almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
differentiable, and its derivative can be used as probability density: \fracF(x) = f(x). 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 In probability theory and statistics, a probability distribution is a Function (mathematics), function that gives the probabilities of occurrence of possible events for an Experiment (probability theory), experiment. It is a mathematical descri ...
precisely if they differ only on a set of Lebesgue
measure zero In mathematical analysis, a null set is a Lebesgue measurable set of real numbers that has Lebesgue measure, measure zero. This can be characterized as a set that can be Cover (topology), covered by a countable union of Interval (mathematics), ...
. In the field of
statistical physics In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. Sometimes called statistical physics or statistical thermodynamics, its applicati ...
, 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: \Pr(t


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 In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line ...
. (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 A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
having the Rademacher distribution—that is, taking −1 or 1 for values, with probability each. The density of probability associated with this variable is: f(t) = \frac (\delta(t+1)+\delta(t-1)). More generally, if a discrete variable can take different values among real numbers, then the associated probability density function is: f(t) = \sum_^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 A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
,
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
, and kurtosis), starting from the formulas given for a continuous distribution of the probability.


Families of densities

It is common for probability density functions (and
probability mass function In probability and statistics, a probability mass function (sometimes called ''probability function'' or ''frequency function'') is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes i ...
s) to be parametrized—that is, to be characterized by unspecified
parameter A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
s. For example, the
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
is parametrized in terms of the
mean A mean is a quantity representing the "center" of a collection of numbers and is intermediate to the extreme values of the set of numbers. There are several kinds of means (or "measures of central tendency") in mathematics, especially in statist ...
and the
variance In probability theory and statistics, variance is the expected value of the squared deviation from the mean of a random variable. The standard deviation (SD) is obtained as the square root of the variance. Variance is a measure of dispersion ...
, denoted by \mu and \sigma^2 respectively, giving the family of densities f(x;\mu,\sigma^2) = \frac e^. Different values of the parameters describe different distributions of different
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
s on the same
sample space In probability theory, the sample space (also called sample description space, possibility space, or outcome space) of an experiment or random trial is the set of all possible outcomes or results of that experiment. A sample space is usually den ...
(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.


Densities associated with multiple variables

For continuous
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
s , 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 \Pr \left( X_1,\ldots,X_n \isin D \right) = \int_D f_(x_1,\ldots,x_n)\,dx_1 \cdots dx_n. If is the
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Ever ...
of the vector , then the joint probability density function can be computed as a partial derivative f(x) = \left.\frac \_x


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: f_(x_i) = \int f(x_1,\ldots,x_n)\, dx_1 \cdots dx_\,dx_\cdots dx_n .


Independence

Continuous random variables admitting a joint density are all independent from each other if f_(x_1,\ldots,x_n) = f_(x_1)\cdots f_(x_n).


Corollary

If the joint probability density function of a vector of random variables can be factored into a product of functions of one variable f_(x_1,\ldots,x_n) = f_1(x_1)\cdots f_n(x_n), (where each is not necessarily a density) 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 f_(x_i) = \frac.


Example

This elementary example illustrates the above definition of multidimensional probability density functions in the simple case of a function of a set of two variables. Let us call \vec R a 2-dimensional random vector of coordinates : the probability to obtain \vec R in the quarter plane of positive and is \Pr \left( X > 0, Y > 0 \right) = \int_0^\infty \int_0^\infty f_(x,y)\,dx\,dy.


Function of random variables and change of variables in the probability density function

If the probability density function of a random variable (or vector) is given as , it is possible (but often not necessary; see below) to calculate the probability density function of some variable . This is also called a "change of variable" and is in practice used to generate a random variable of arbitrary shape using a known (for instance, uniform) random number generator. It is tempting to think that in order to find the expected value , one must first find the probability density of the new random variable . However, rather than computing \operatorname E\big(g(X)\big) = \int_^\infty y f_(y)\,dy, one may find instead \operatorname E\big(g(X)\big) = \int_^\infty g(x) f_X(x)\,dx. The values of the two integrals are the same in all cases in which both and actually have probability density functions. It is not necessary that be a
one-to-one function In mathematics, an injective function (also known as injection, or one-to-one function ) is a function that maps distinct elements of its domain to distinct elements of its codomain; that is, implies (equivalently by contraposition, impl ...
. In some cases the latter integral is computed much more easily than the former. See Law of the unconscious statistician.


Scalar to scalar

Let g: \Reals \to \Reals be a
monotonic function In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of or ...
, then the resulting density function is f_Y(y) = f_X\big(g^(y)\big) \left, \frac \big(g^(y)\big) \. Here denotes the
inverse function In mathematics, the inverse function of a function (also called the inverse of ) is a function that undoes the operation of . The inverse of exists if and only if is bijective, and if it exists, is denoted by f^ . For a function f\colon ...
. This follows from the fact that the probability contained in a differential area must be invariant under change of variables. That is, \left, f_Y(y)\, dy \ = \left, f_X(x)\, dx \, or f_Y(y) = \left, \frac \ f_X(x) = \left, \frac (x) \ f_X(x) = \left, \frac \big(g^(y)\big) \ f_X\big(g^(y)\big) = \cdot f_X\big(g^(y)\big) . For functions that are not monotonic, the probability density function for is \sum_^ \left, \frac g^_(y) \ \cdot f_X\big(g^_(y)\big), where is the number of solutions in for the equation g(x) = y, and g_k^(y) are these solutions.


Vector to vector

Suppose is an -dimensional random variable with joint density . If , where is a
bijective In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
,
differentiable function In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
, then has density : p_(\mathbf) = f\Bigl(G^(\mathbf)\Bigr) \left, \det\left _\right\ with the differential regarded as the Jacobian of the inverse of , evaluated at . For example, in the 2-dimensional case , suppose the transform is given as , with inverses , . The joint distribution for y = (''y''1, y2) has density p_(y_1,y_2) = f_\big(G_1^(y_1,y_2), G_2^(y_1,y_2)\big) \left\vert \frac \frac - \frac \frac \right\vert.


Vector to scalar

Let V: \R^n \to \R be a differentiable function and X be a random vector taking values in \R^n , f_X be the probability density function of X and \delta(\cdot) be the Dirac delta function. It is possible to use the formulas above to determine f_Y , the probability density function of Y = V(X) , which will be given by f_Y(y) = \int_ f_(\mathbf) \delta\big(y - V(\mathbf)\big) \,d \mathbf. This result leads to the law of the unconscious statistician: \begin \operatorname_Y &=\int_ y f_Y(y) \, dy \\ &= \int_ y \int_ f_X(\mathbf) \delta\big(y - V(\mathbf)\big) \,d \mathbf \,dy \\ &= \int_ \int_ y f_(\mathbf) \delta\big(y - V(\mathbf)\big) \, dy \, d \mathbf \\ &= \int_ V(\mathbf) f_X(\mathbf) \, d \mathbf=\operatorname_X (X) \end ''Proof:'' Let Z be a collapsed random variable with probability density function p_Z(z) = \delta(z) (i.e., a constant equal to zero). Let the random vector \tilde and the transform H be defined as H(Z,X)=\begin Z+V(X)\\ X\end=\begin Y\\ \tilde\end. It is clear that H is a bijective mapping, and the Jacobian of H^ is given by: \frac=\begin 1 & -\frac\\ \mathbf_ & \mathbf_ \end, which is an upper triangular matrix with ones on the main diagonal, therefore its determinant is 1. Applying the change of variable theorem from the previous section we obtain that f_(y,x) = f_X(\mathbf) \delta\big(y - V(\mathbf)\big), which if marginalized over x leads to the desired probability density function.


Sums of independent random variables

The probability density function of the sum of two independent random variables and , each of which has a probability density function, is the
convolution In mathematics (in particular, functional analysis), convolution is a operation (mathematics), mathematical operation on two function (mathematics), functions f and g that produces a third function f*g, as the integral of the product of the two ...
of their separate density functions: f_(x) = \int_^\infty f_U(y) f_V(x - y)\,dy = \left( f_ * f_ \right) (x) It is possible to generalize the previous relation to a sum of N independent random variables, with densities : f_(x) = \left( f_ * \cdots * f_ \right) (x) This can be derived from a two-way change of variables involving and , similarly to the example below for the quotient of independent random variables.


Products and quotients of independent random variables

Given two independent random variables and , each of which has a probability density function, the density of the product and quotient can be computed by a change of variables.


Example: Quotient distribution

To compute the quotient of two independent random variables and , define the following transformation: \begin Y &= U/V \\ exZ &= V \end Then, the joint density can be computed by a change of variables from ''U'',''V'' to ''Y'',''Z'', and can be derived by marginalizing out from the joint density. The inverse transformation is \begin U &= YZ \\ V &= Z \end The absolute value of the
Jacobian matrix In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of component ...
determinant J(U,V\mid Y,Z) of this transformation is: \left, \det\begin \frac & \frac \\ \frac & \frac \end \ = \left, \det\begin z & y \\ 0 & 1 \end \ = , z, . Thus: p(y,z) = p(u,v)\,J(u,v\mid y,z) = p(u)\,p(v)\,J(u,v\mid y,z) = p_U(yz)\,p_V(z)\, , z, . And the distribution of can be computed by marginalizing out : p(y) = \int_^\infty p_U(yz)\,p_V(z)\, , z, \, dz This method crucially requires that the transformation from ''U'',''V'' to ''Y'',''Z'' be
bijective In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equival ...
. The above transformation meets this because can be mapped directly back to , and for a given the quotient is
monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of ord ...
. This is similarly the case for the sum , difference and product . Exactly the same method can be used to compute the distribution of other functions of multiple independent random variables.


Example: Quotient of two standard normals

Given two standard normal variables and , the quotient can be computed as follows. First, the variables have the following density functions: \begin p(u) &= \frac e^ \\ exp(v) &= \frac e^ \end We transform as described above: \begin Y &= U/V \\ exZ &= V \end This leads to: \begin p(y) &= \int_^\infty p_U(yz)\,p_V(z)\, , z, \, dz \\ pt&= \int_^\infty \frac e^ \frac e^ , z, \, dz \\ pt&= \int_^\infty \frac e^ , z, \, dz \\ pt&= 2\int_0^\infty \frac e^ z \, dz \\ pt&= \int_0^\infty \frac e^ \, du && u=\tfracz^2\\ pt&= \left. -\frac e^\_^\infty \\ pt&= \frac \end This is the density of a standard Cauchy distribution.


See also

* * * * * * * * Merging independent probability density functions * Uses as ''position probability density'': ** **


References


Further reading

* * * Chapters 7 to 9 are about continuous variables.


External links

* * {{DEFAULTSORT:Probability Density Function Functions related to probability distributions Equations of physics Probability theory