In
probability theory
Probability theory 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 expressing it through a set o ...
, a probability density function (PDF), or density of a
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 close 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.
In a more precise sense, 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 mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
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 assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
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 the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
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
In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
(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 4 to 6 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 dying 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 nanoseconds = 1 hour).
There is a probability density function ''f'' with ''f''(5 hours) = 2 hour
−1. The
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with ...
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 mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
has density
, where
is a non-negative
Lebesgue-integrable
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis. The Lebesgue integral, named after French mathematician Henri Leb ...
function, if:
Hence, if
is the
cumulative distribution function of
, then:
and (if
is continuous at
)
Intuitively, one can think of
as being the probability of
falling within the infinitesimal
interval
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 mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
with values in a
measurable space (usually
with the
Borel sets as measurable subsets) has as
probability distribution
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
the 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
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 ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
. The
probability mass function
In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
of a
discrete random variable is the density with respect to the
counting measure over the sample space (usually the set of
integers, or some subset thereof).
It is not possible to define a density with reference to an arbitrary measure (e.g. one can't 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.
Further details
Unlike a probability, a probability density function can take on values greater than one; for example, the uniform distribution on the interval
, 1/2has probability density f(x) = 2 for 0 ≤ x ≤ 1/2 and f(x) = 0 elsewhere.
The standard
normal distribution
In 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 e^
The parameter \mu i ...
has probability density
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, mathematical expectation, mean, average, or first moment) is a generalization of the weighted average. Informally, the expected value is the arithmetic mean of a l ...
of (if the expected value exists) 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
The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function.
This distribution has neither a probability density function nor a probability mass function, since although its cumulative ...
, 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 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
In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...
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
In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
. (This is not possible with a probability density function in the sense defined above, it may be done with a
distribution Distribution may refer to:
Mathematics
*Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations
*Probability distribution, the probability of a particular value or value range of a varia ...
.) For example, consider a binary 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. It is a mapping or a function from possible outcomes (e.g., the po ...
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 (such as the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the '' ari ...
,
variance, 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 is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
s) to be parametrized—that is, to be characterized by unspecified
parameters. For example, the
normal distribution
In 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 e^
The parameter \mu i ...
is parametrized in terms of the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the '' ari ...
and the
variance, denoted by
and
respectively, giving the family of densities
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 mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s 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.
Densities associated with multiple variables
For continuous
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. It is a mapping or a function from possible outcomes (e.g., the po ...
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
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 and only 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
(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
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
a 2-dimensional random vector of coordinates : the probability to obtain
in the quarter plane of positive and is
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
one may find instead
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 some cases the latter integral is computed much more easily than the former. See
Law of the unconscious statistician.
Scalar to scalar
Let
be a
monotonic function, then the resulting density function is
Here denotes the
inverse function.
This follows from the fact that the probability contained in a differential area must be invariant under change of variables. That is,
or
For functions that are not monotonic, the probability density function for is
where is the number of solutions in for the equation
, and
are these solutions.
Vector to vector
Suppose is an -dimensional random variable with joint density . If , where is a
bijective,
differentiable function, then has density :
with the differential regarded as the
Jacobian
In mathematics, a Jacobian, named for Carl Gustav Jacob Jacobi, may refer to:
*Jacobian matrix and determinant
*Jacobian elliptic functions
*Jacobian variety
*Intermediate Jacobian
In mathematics, the intermediate Jacobian of a compact Kähler m ...
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, y
2) has density
Vector to scalar
Let
be a differentiable function and
be a random vector taking values in
,
be the probability density function of
and
be the
Dirac delta function. It is possible to use the formulas above to determine
, the probability density function of
, which will be given by
This result leads to the
law of the unconscious statistician:
''Proof:''
Let
be a collapsed random variable with probability density function
(i.e., a constant equal to zero). Let the random vector
and the transform
be defined as
:
It is clear that
is a bijective mapping, and the Jacobian of
is given by:
:
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
:
which if marginalized over
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 of their separate density functions:
It is possible to generalize the previous relation to a sum of N independent random variables, with densities :
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 ''U'' and ''V'', 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:
Then, the joint density can be computed by a change of variables from ''U'',''V'' to ''Y'',''Z'', and ''Y'' can be derived by
marginalizing out
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the varia ...
''Z'' from the joint density.
The inverse transformation is
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. When this matrix is square, that is, when the function takes the same number of variables as ...
determinant
of this transformation is:
Thus:
And the distribution of ''Y'' can be computed by
marginalizing out
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the varia ...
''Z'':
This method crucially requires that the transformation from ''U'',''V'' to ''Y'',''Z'' be
bijective. The above transformation meets this because ''Z'' can be mapped directly back to ''V'', and for a given ''V'' the quotient is
monotonic. 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 ''U'' and ''V'', the quotient can be computed as follows. First, the variables have the following density functions:
We transform as described above:
This leads to:
This is the density of a standard
Cauchy distribution
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution. It is also known, especially among physicists, as the Lorentz distribution (after Hendrik Lorentz), Cauchy–Lorentz distribution, Lorentz(ian) fu ...
.
See also
*
Density estimation
*
Kernel density estimation
In statistics, kernel density estimation (KDE) is the application of kernel smoothing for probability density estimation, i.e., a non-parametric method to estimate the probability density function of a random variable based on '' kernels'' as ...
*
Likelihood function
*
List of probability distributions
*
Probability amplitude
*
Probability mass function
In probability and statistics, a probability mass function is a function that gives the probability that a discrete random variable is exactly equal to some value. Sometimes it is also known as the discrete density function. The probability mass ...
*
Secondary measure
* Uses as ''position probability density'':
**
Atomic orbital
In atomic theory and quantum mechanics, an atomic orbital is a function describing the location and wave-like behavior of an electron in an atom. This function can be used to calculate the probability of finding any electron of an atom in an ...
**
Home range
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