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Encyclopedia > Probability density function

In mathematics, a probability density function (pdf) is a function that represents a probability distribution in terms of integrals. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... In calculus, the integral of a function is an extension of the concept of a sum. ...


Formally, a probability distribution has density f if f is a non-negative Lebesgue-integrable function such that the probability of the interval [a, b] is given by The integral can be interpreted as the area under a curve. ...

for any two numbers a and b. This implies that the total integral of f must be 1. Conversely, any non-negative Lebesgue-integrable function with total integral 1 is the probability density of a suitably defined probability distribution.


Intuitively, if a probability distribution has density f(x), then the infinitesimal interval [x, x + dx] has probability f(x) dx. In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ...


Informally, a probability density function can be seen as a "smoothed out" version of a histogram: if one empirically samples enough values of a continuous random variable, producing a histogram depicting relative frequencies of output ranges, then this histogram will resemble the random variable's probability density, assuming that the output ranges are sufficiently narrow. Example of a histogram of 100 normally distributed random values. ... By one convention, a random variable X is called continuous if its cumulative distribution function is continuous. ...

Contents

Simplified explanation

A probability density function is any function f(x) that describes the probability density in terms of the input variable x in a manner described below.

  • f(x) is greater than or equal to zero for all values of x
  • The total area under the graph is 1:

The actual probability can then be calculated by taking the integral of the function f(x) by the integration interval of the input variable x.


For example: the variable x being within the interval [4.3,7.8] would have the actual probability of

Further details

For example, the continuous uniform distribution on the interval [0,1] has probability density f(x) = 1 for 0 ≤ x ≤ 1 and f(x) = 0 elsewhere. The standard normal distribution has probability density In probability theory and statistics, the continuous uniform distribution is a family of probability distributions such that for each member of the family, all intervals of the same length on the distributions support are equally probable. ... The normal distribution, also called Gaussian distribution by scientists (named after Carl Friedrich Gauss due to his rigorous application of the distribution to astronomical data (Havil, 2003)), is a continuous probability distribution of great importance in many fields. ...

If a random variable X is given and its distribution admits a probability density function f(x), then the expected value of X (if it exists) can be calculated as In probability theory the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects as the outcome of the random trial when identical odds are...

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. In mathematics, a random variable is discrete if its probability distribution is discrete; a discrete probability distribution is one that is fully characterized by a probability mass function. ... The Cantor distribution is the probability distribution whose cumulative distribution function is the Cantor function. ...


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: In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than... // Absolute continuity of real functions In mathematics, a real-valued function f of a real variable is absolutely continuous on a specified finite or infinite interval if for every positive number ε, no matter how small, there is a positive number δ small enough so that whenever a sequence of pairwise disjoint... In measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. ... For a non-technical overview of the subject, see Calculus. ...

If a probability distribution admits a density, then the probability of every one-point set {a} is zero.


It is a common mistake to think of f(x) as the probability of {x}, but this is incorrect; in fact, f(x) will often be bigger than 1 - consider a random variable that is uniformly distributed between 0 and ½. Loosely, one may think of f(xdx as the probability that a random variable whose probability density function if f is in the interval from x to x + dx, where dx is an infinitely small increment. In mathematics, the uniform distributions are simple probability distributions. ...


Two probability densities f and g represent the same probability distribution precisely if they differ only on a set of Lebesgue measure zero. In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... In mathematics, the Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. ... Let μ be a measure on a sigma algebra Σ of subsets of a set X. An element A in Σ is said to have measure zero if μ(A)=0. ...


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: Statistical physics, one of the fundamental theories of physics, uses methods of statistics in solving physical problems. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than...


If dt is an infinitely small number, the probability that X is included within the interval (tt + dt) is equal to , or:

Link between discrete and continuous distributions

The definition of a probability density function at the start of this page makes it possible to describe the variable associated with a continuous distribution using a set of binary discrete variables associated with the intervals [ab] (for example, a variable being worth 1 if X is in [ab], and 0 if not).


It is also possible to represent certain discrete random variables using a density of probability, via the Dirac delta function. For example, let us consider a binary discrete random variable taking −1 or 1 for values, with probability ½ each. The Dirac delta function, often referred to as the unit impulse function and introduced by the British theoretical physicist Paul Dirac, can usually be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere. ... A random variable is a mathematical function that maps outcomes of random experiments to numbers. ...


The density of probability associated with this variable is:

More generally, if a discrete variable can take 'n' 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 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. In statistics, mean has two related meanings: Look up mean in Wiktionary, the free dictionary. ... In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ... The far red light has no effect on the average speed of the gravitropic reaction in wheat coleoptiles, but it changes kurtosis from platykurtic to leptokurtic (-0. ...


In physics, this description is also useful in order to characterize mathematically the initial configuration of a Brownian movement. Physics (Greek: (phúsis), nature and (phusiké), knowledge of nature) is the science concerned with the discovery and characterization of universal laws which govern matter, energy, space, and time. ... An example of 1000 steps of Brownian motion in two dimensions. ...


Probability function associated to 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 n variables, such that, for any domain D in the n-dimensional space of the values of the variables , the probability that a realisation of the set variables falls inside the domain D is A random variable is a mathematical function that maps outcomes of random experiments to numbers. ...

For i=1, 2, …,n, let be the probability density function associated to variable Xi alone. This probability density can be deduced from the probability densities associated of the random variables by integrating on all values of the n − 1 other variables:

Independence

Continuous random variables are all independent from each other if and only if

Corollary

If the joint probability density function of a vector of n random variables can be factored into a product of n functions of one variable

then the n 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 (X,Y): the probability to obtain in the quarter plane of positive x and y is

Sums of independent random variables

The probability density function of the sum of two independent random variables U and V, each of which has a probability density function is the convolution of their separate density functions: In mathematics and, in particular, functional analysis, convolution is a mathematical operator which takes two functions f and g and produces a third function that in a sense represents the amount of overlap between f and a reversed and translated version of g. ...

Dependent variables

If the probability density function of an independent random variable x is given as f(x), it is possible (but often not necessary; see below) to calculate the probability density function of some variable y which depends on x. If the dependence is y = g(x) and the function g is monotonic, then the resulting density function is In mathematics, functions between ordered sets are monotonic (or monotone, or even isotone) if they preserve the given order. ...

Here g−1 denotes the inverse function and g' denotes the derivative. In mathematics, an inverse function is in simple terms a function which does the reverse of a given function. ... For a non-technical overview of the subject, see Calculus. ...


For functions which are not monotonic the probability density function for y is

where n(y) is the number of solutions in x for the equation g(x) = y, and are these solutions.


It is tempting to think that in order to find the expected value E(g(X)) one must first find the probability density of g(X). However, rather than computing

one may find instead

The values of the two integrals are the same in all cases in which both X and g(X) actually have probability density functions. It is not necessary that g be a one-to-one function. In some cases the latter integral is computed much more easily than the former. In mathematics, an injection, surjection or bijection is a certain type of function. ...


Multiple variables

The above formulas can be generalized to variables (which we will again call y) depending on more than one other variables. f(x0, x1, ..., xm-1) shall denote the probability density function of the variables y depends on, and the dependence shall be y = g(x0, x1, ..., xm-1). Then, the resulting density function is

where the integral is over the entire (m-1)-dimensional solution of the subscripted equation and the symbolic dV must be replaced by a parametrization of this solution for a particular calculation; the variables x0, x1, ..., xm-1 are then of course functions of this parametrization.


Finding moments

In particular, the nth moment E(Xn) of the probability distribution of a random variable X is given by-1...

and the variance is In probability theory and statistics, the variance of a random variable (or somewhat more precisely, of a probability distribution) is a measure of its statistical dispersion, indicating how its possible values are spread around the expected value. ...

or, expanding, gives:

.

See also


  Results from FactBites:
 
Probability density function - Wikipedia, the free encyclopedia (898 words)
A probability density function is non-negative everywhere and its integral from −∞ to +∞ is equal to 1.
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.
The definition of a probability density function at the start of this page makes it possible to describe the variable associated with a continuous distribution using a set of binary discrete variables associated with the intervals [a; b] (for example, a variable being worth 1 if X is in [a; b], and 0 if not).
  More results at FactBites »

 
 

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