# Probability density function

**probability densitydensity functiondensityPDFjoint probability density functionprobability densitiesdistributionprobability density function (PDF)densitiesdensity functions**

In probability theory, 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 equal that sample.wikipedia

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### Random variable

**random variablesrandom variationrandom**

In a more precise sense, 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. A random variable X with values in a measurable space (usually with the Borel sets as measurable subsets) has as probability distribution the measure X ∗ P on : the density of X with respect to a reference measure \mu on is the Radon–Nikodym derivative:

Random variables can be discrete, that is, taking any of a specified finite or countable list of values, endowed with a probability mass function characteristic of the random variable's probability distribution; or continuous, taking any numerical value in an interval or collection of intervals, via a probability density function that is characteristic of the random variable's probability distribution; or a mixture of both types.

### Cumulative distribution function

**distribution functionCDFcumulative probability distribution function**

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. A distribution has a density function if and only if its cumulative distribution function F(x) is absolutely continuous.

In the case of a scalar continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables.

### Probability distribution

**distributioncontinuous probability distributiondiscrete probability distribution**

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. In probability theory, 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 equal that sample. A random variable X with values in a measurable space (usually with the Borel sets as measurable subsets) has as probability distribution the measure X ∗ P on : the density of X with respect to a reference measure \mu on is the Radon–Nikodym derivative:

On the other hand, a continuous probability distribution (applicable to the scenarios where the set of possible outcomes can take on values in a continuous range (e.g. real numbers), such as the temperature on a given day) is typically described by probability density functions (with the probability of any individual outcome actually being 0).

### Integral

**integrationintegral calculusdefinite integral**

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.

Moreover, the integral under an entire probability density function must equal 1, which provides a test of whether a function with no negative values could be a density function or not.

### Probability mass function

**mass functionprobability massmass**

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.

A probability mass function differs from a probability density function (PDF) in that the latter is associated with continuous rather than discrete random variables.

### Uniform distribution (continuous)

**uniform distributionuniformuniformly distributed**

Unlike a probability, a probability density function can take on values greater than one; for example, the uniform distribution on the interval [0, ½] has probability density f(x) = 2 for 0 ≤ x ≤ ½ and f(x) = 0 elsewhere.

The probability density function of the continuous uniform distribution is:

### Expected value

**expectationexpectedmean**

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

The same principle applies to an absolutely continuous random variable, except that an integral of the variable with respect to its probability density replaces the sum.

### Cantor distribution

**CantorCantor measureneither**

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.

This distribution has neither a probability density function nor a probability mass function, since although its cumulative distribution function is a continuous function, the distribution is not absolutely continuous with respect to Lebesgue measure, nor does it have any point-masses.

### Radon–Nikodym theorem

**Radon–Nikodym derivativeRadon-Nikodym theoremdensity**

A random variable X with values in a measurable space (usually with the Borel sets as measurable subsets) has as probability distribution the measure X ∗ P on : the density of X with respect to a reference measure \mu on is the Radon–Nikodym derivative:

Specifically, the probability density function of a random variable is the Radon–Nikodym derivative of the induced measure with respect to some base measure (usually the Lebesgue measure for continuous random variables).

### Variance

**sample variancepopulation variancevariability**

For instance, the above expression allows for determining statistical characteristics of such a discrete variable (such as its mean, its variance and its kurtosis), starting from the formulas given for a continuous distribution of the probability.

If the random variable X has a probability density function f(x), and F(x) is the corresponding cumulative distribution function, then

### Kernel (statistics)

**kernelkernelskernel function**

This normalization factor is outside the kernel of the distribution.

In statistics, especially in Bayesian statistics, the kernel of a probability density function (pdf) or probability mass function (pmf) is the form of the pdf or pmf in which any factors that are not functions of any of the variables in the domain are omitted.

### Normalizing constant

**normalizednormalization constantnormalization**

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).

In probability theory, a normalizing constant is a constant by which an everywhere non-negative function must be multiplied so the area under its graph is 1, e.g., to make it a probability density function or a probability mass function.

### Mean

**mean valueaveragepopulation mean**

For instance, the above expression allows for determining statistical characteristics of such a discrete variable (such as its mean, its variance and its kurtosis), starting from the formulas given for a continuous distribution of the probability.

For a continuous distribution,the mean is, where f(x) is the probability density function.

### Absolute continuity

**absolutely continuousabsolutely continuous measureAbsolutely continuous function**

A distribution has a density function if and only if its cumulative distribution function F(x) is absolutely continuous.

Thus, the absolutely continuous measures on R n are precisely those that have densities; as a special case, the absolutely continuous probability measures are precisely the ones that have probability density functions.

### Normal distribution

**normally distributedGaussian distributionnormal**

The standard normal distribution has probability density

The probability density of the normal distribution is

### Kernel density estimation

**Parzen windowkernel densitykernel**

In statistics, kernel density estimation (KDE) is a non-parametric way to estimate the probability density function of a random variable.

### Independence (probability theory)

**independentstatistically independentindependence**

Continuous random variables X 1, ..., X n admitting a joint density are all independent from each other if and only if

or equivalently, if the probability densities f_X(x) and f_Y(y) and the joint probability density exist,

### Law of the unconscious statistician

See Law of the unconscious statistician.

If it is a continuous distribution and one knows its probability density function ƒ X (but not ƒ g(X) ), then the expected value of g(X) is

### Atomic orbital

**orbitalatomic orbitalselectron cloud**

functions that show probability density more directly, see pictures of d-orbitals below.]]

### Density estimation

**estimatingdensity estimatesestimate**

density estimation is the construction of an estimate, based on observed data, of an unobservable underlying probability density function.

### Kurtosis

**excess kurtosisleptokurticplatykurtic**

For instance, the above expression allows for determining statistical characteristics of such a discrete variable (such as its mean, its variance and its kurtosis), starting from the formulas given for a continuous distribution of the probability.

The probability density function is given by

### Cauchy distribution

**LorentzianCauchyLorentzian distribution**

This is the density of a standard Cauchy distribution.

The Cauchy distribution has the probability density function (PDF)

### Likelihood function

**likelihoodlikelihood ratiolog-likelihood**

Let X be a random variable following an absolutely continuous probability distribution with density function f depending on a parameter \theta.

### Dirac delta function

**Dirac deltadelta functionimpulse**

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, by using the Dirac delta function.

It represents the probability density at time t = ε of the position of a particle starting at the origin following a standard Brownian motion.

### Secondary measure

When ρ is a probability density function, a sufficient condition so that μ, while admitting moments of any order can be a secondary measure associated with ρ is that its Stieltjes Transformation is given by an equality of the type: