{{Use American English|date = January 2019}}{{Short description|Function whose integral over a region describes the probability of an event occurring in that region}} Suppose bacteria of a certain species typically live 20 to 30 hours. The probability that a bacterium lives {{em|exactly}} 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 0.087, 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 0.0087, 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) ≈ {{val|6e-13}} (using the unit conversion {{val|3.6e12}} nanoseconds = 1 hour).There is a probability density function f with f(5 hours) = 2 hour−1. The integral 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 X has density f_X, where f_X is a non-negative Lebesgue-integrable function, if: Hence, if F_X is the cumulative distribution function of X, then:F_X(x) = int_{-infty}^x f_X(u) , du ,and (if f_X is continuous at x) Intuitively, one can think of f_X(x) , dx as being the probability of X falling within the infinitesimal interval [x,x+dx].

Formal definition

(This definition may be extended to any probability distribution using the measure-theoretic definition of probability.)A random variable X with values in a measurable space (mathcal{X}, mathcal{A}) (usually mathbb{R}^n with the Borel sets as measurable subsets) has as probability distribution the measure X∗P on (mathcal{X}, mathcal{A}): the density of X with respect to a reference measure mu on (mathcal{X}, mathcal{A}) is the Radon–Nikodym derivative:f = frac{dX_*P}{dmu} .That is, f is any measurable function with the property that:Pr [X in A ] = int_{X^{-1} A} , dP = int_A f , dmu for any measurable set A in mathcal{A}.

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 (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 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.

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 {{closed-closed|0, 1/2}} has probability density {{math|1=f(x) = 2}} for {{math|0 ≤ x ≤ 1/2}} and {{math|1=f(x) = 0}} elsewhere.The standard normal distribution has probability densityf(x) = frac{1}{sqrt{2pi}}, e^{-x^2/2}.If a random variable {{math|X}} is given and its distribution admits a probability density function {{math|f}}, then the expected value of {{math|X}} (if the expected value exists) can be calculated asoperatorname{E}[X] = int_{-infty}^infty x,f(x),dx.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 and only if its cumulative distribution function {{math|F(x)}} is absolutely continuous. In this case: {{math|F}} is almost everywhere differentiable, and its derivative can be used as probability density:frac{d}{dx}F(x) = f(x).If a probability distribution admits a density, then the probability of every one-point set {{math|{a}}} is zero; the same holds for finite and countable sets.Two probability densities {{math|f}} and {{math|g}} 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 {{math|dt}} is an infinitely small number, the probability that {{math|X}} is included within the interval {{open-open|t, t + dt}} is equal to {{math|f(t) dt}}, or:Pr(tFor continuous random variables {{math|X 0 right)

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

If the probability density function of a random variable (or vector) {{math|X}} is given as {{math|fX(x)}}, it is possible (but often not necessary; see below) to calculate the probability density function of some variable {{math|1=Y = g(X)}}. This is also called a “change of variable” and is in practice used to generate a random variable of arbitrary shape {{math|1=fg(X) = fY}} using a known (for instance, uniform) random number generator.It is tempting to think that in order to find the expected value {{math|E(g(X))}}, one must first find the probability density {{math|fg(X)}} of the new random variable {{math|1=Y = g(X)}}. However, rather than computingoperatorname Ebig(g(X)big) = int_{-infty}^infty y f_{g(X)}(y),dy, one may find insteadoperatorname Ebig(g(X)big) = int_{-infty}^infty g(x) f_X(x),dx.The values of the two integrals are the same in all cases in which both {{math|X}} and {{math|g(X)}} actually have probability density functions. It is not necessary that {{math|g}} 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 g: Reals to Reals be a monotonic function, then the resulting density function isWEB, Siegrist, Kyle, Transformations of Random Variables,stats.libretexts.org/Bookshelves/Probability_Theory/Probability_Mathematical_Statistics_and_Stochastic_Processes_%28Siegrist%29/03%3A_Distributions/3.07%3A_Transformations_of_Random_Variables#The_Change_of_Variables_Formula, LibreTexts Statistics, 22 December 2023, f_Y(y) = f_Xbig(g^{-1}(y)big) left| frac{d}{dy} big(g^{-1}(y)big) right|.Here {{math|g−1}} 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,left| f_Y(y), dy right| = left| f_X(x), dx right|,orf_Y(y) = left| frac{dx}{dy} right| f_X(x)

left| frac{d}{dy} (x) right| f_X(x)

left| frac{d}{dy} big(g^{-1}(y)big) right| f_Xbig(g^{-1}(y)big)

{left|left(g^{-1}right)’(y)right|} cdot f_Xbig(g^{-1}(y)big) .

For functions that are not monotonic, the probability density function for {{mvar|y}} issum_{k=1}^{n(y)} left| frac{d}{dy} g^{-1}_{k}(y) right| cdot f_Xbig(g^{-1}_{k}(y)big),where {{math|n(y)}} is the number of solutions in {{mvar|x}} for the equation g(x) = y, and g_k^{-1}(y) are these solutions.

Vector to vector

Suppose {{math|x}} is an {{mvar|n}}-dimensional random variable with joint density {{math|f}}. If {{math|1=y = G(x)}}, where {{math|G}} is a bijective, differentiable function, then {{math|y}} has density {{math|}}: with the differential regarded as the Jacobian of the inverse of {{math|G(â‹…)}}, evaluated at {{math|y}}.BOOK, Jay L., Devore, Kenneth N., Berk, Modern Mathematical Statistics with Applications, Cengage, 2007, 978-0-534-40473-4, 263,books.google.com/books?id=3X7Qca6CcfkC&pg=PA263, For example, in the 2-dimensional case {{math|1=x = (x1, x2)}}, suppose the transform {{math|G}} is given as {{math|1=y1 = G1(x1, x2)}}, {{math|1=y2 = G2(x1, x2)}} with inverses {{math|1=x1 = G1−1(y1, y2)}}, {{math|1=x2 = G2−1(y1, y2)}}. The joint distribution for y = (y1, y2) has densityBOOK, Elementary Probability, David, Stirzaker, 2007-01-01, Cambridge University Press, 978-0521534284, 851313783, p_{Y_1, Y_2}(y_1,y_2) = f_{X_1,X_2}big(G_1^{-1}(y_1,y_2), G_2^{-1}(y_1,y_2)big) leftvert frac{partial G_1^{-1}}{partial y_1} frac{partial G_2^{-1}}{partial y_2} - frac{partial G_1^{-1}}{partial y_2} frac{partial G_2^{-1}}{partial y_1} rightvert.

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 byf_Y(y) = int_{R^n} f_{X}(mathbf{x}) deltabig(y - V(mathbf{x})big) ,d mathbf{x}.This result leads to the law of the unconscious statistician:operatorname{E}_Y[Y] =int_{R} y f_Y(y) , dy = int_{R} y int_{R^n} f_X(mathbf{x}) deltabig(y - V(mathbf{x})big) ,d mathbf{x} ,dy = int_{{mathbb R}^n} int_{mathbb R} y f_{X}(mathbf{x}) deltabig(y - V(mathbf{x})big) , dy , d mathbf{x}= int_{mathbb R^n} V(mathbf{x}) f_X(mathbf{x}) , d mathbf{x}=operatorname{E}_X[V(X)].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{X} and the transform H be defined asH(Z,X)=begin{bmatrix} Z+V(X) Xend{bmatrix}=begin{bmatrix} Y tilde{X}end{bmatrix}.It is clear that H is a bijective mapping, and the Jacobian of H^{-1} is given by:frac{dH^{-1}(y,tilde{mathbf{x}})}{dy,dtilde{mathbf{x}}}=begin{bmatrix} 1 & -frac{dV(tilde{mathbf{x}})}{dtilde{mathbf{x}}} mathbf{0}_{ntimes1} & mathbf{I}_{ntimes n} end{bmatrix},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 thatf_{Y,X}(y,x) = f_X(mathbf{x}) deltabig(y - V(mathbf{x})big),which if marginalized over x leads to the desired probability density function.

Sums of independent random variables

{{See also|List of convolutions of probability distributions}}The probability density function of the sum of two independent random variables {{math|U}} and {{math|V}}, each of which has a probability density function, is the convolution of their separate density functions:f_{U+V}(x) = int_{-infty}^x f_U(y) f_V(x - y),dy

left( f_{U} * f_{V} right) (x)

It is possible to generalize the previous relation to a sum of N independent random variables, with densities {{math|U1, ..., UN}}:f_{U_1 + cdots + U}(x)

left( f_{U_1} * cdots * f_{U_N} right) (x)

This can be derived from a two-way change of variables involving {{math|1=Y = U + V}} and {{math|1=Z = V}}, similarly to the example below for the quotient of independent random variables.

Products and quotients of independent random variables

{{See also|Product distribution|Ratio distribution}}Given two independent random variables {{math|U}} and {{math|V}}, each of which has a probability density function, the density of the product {{math|1=Y = UV}} and quotient {{math|1=Y = U/V}} can be computed by a change of variables.

Example: Quotient distribution

To compute the quotient {{math|1=Y = U/V}} of two independent random variables {{math|U}} and {{math|V}}, define the following transformation:begin{align}Y &= U/V [1ex]Z &= Vend{align}Then, the joint density {{math|p(y,z)}} can be computed by a change of variables from U,V to Y,Z, and {{math|Y}} can be derived by marginalizing out {{math|Z}} from the joint density.The inverse transformation isbegin{align}U &= YZ V &= Zend{align}The absolute value of the Jacobian matrix determinant J(U,Vmid Y,Z) of this transformation is:left| detbegin{bmatrix}frac{partial u}{partial y} & frac{partial u}{partial z} frac{partial v}{partial y} & frac{partial v}{partial z}end{bmatrix} right|

left| detbegin{bmatrix}z & y end{bmatrix} right|

|z| .

Thus:p(y,z) = p(u,v),J(u,vmid y,z) = p(u),p(v),J(u,vmid y,z) = p_U(yz),p_V(z), |z| .And the distribution of {{math|Y}} can be computed by marginalizing out {{math|Z}}:p(y) = int_{-infty}^infty p_U(yz),p_V(z), |z| , dzThis method crucially requires that the transformation from U,V to Y,Z be bijective. The above transformation meets this because {{math|Z}} can be mapped directly back to {{math|V}}, and for a given {{math|V}} the quotient {{math|U/V}} is monotonic. This is similarly the case for the sum {{math|U + V}}, difference {{math|U − V}} and product {{math|UV}}.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 {{math|U}} and {{math|V}}, the quotient can be computed as follows. First, the variables have the following density functions:begin{align}p(u) &= frac{1}{sqrt{2pi}} e^{-{u^2}/{2}} [1ex]p(v) &= frac{1}{sqrt{2pi}} e^{-{v^2}/{2}}end{align}We transform as described above:begin{align}Y &= U/V [1ex]Z &= Vend{align}This leads to:begin{align}p(y) &= int_{-infty}^infty p_U(yz),p_V(z), |z| , dz [5pt]&= int_{-infty}^infty frac{1}{sqrt{2pi}} e^{-frac{1}{2} y^2 z^2} frac{1}{sqrt{2pi}} e^{-frac{1}{2} z^2} |z| , dz [5pt]&= int_{-infty}^infty frac{1}{2pi} e^{-frac{1}{2}left(y^2+1right)z^2} |z| , dz [5pt]&= 2int_0^infty frac{1}{2pi} e^{-frac{1}{2}left(y^2+1right)z^2} z , dz [5pt]&= int_0^infty frac{1}{pi} e^{-left(y^2+1right)u} , du && u=tfrac{1}{2}z^2[5pt]&= left. -frac{1}{pi left(y^2+1right)} e^{-left(y^2+1right)u}right|_{u=0}^infty [5pt]&= frac{1}{pi left(y^2+1right)}end{align}This is the density of a standard Cauchy distribution.

See also

• {{Annotated link|Density estimation}}
• {{Annotated link|Kernel density estimation}}
• {{Annotated link|Likelihood function}}
• {{Annotated link|List of probability distributions}}
• {{Annotated link|Probability amplitude}}
• {{Annotated link|Probability mass function}}
• {{Annotated link|Secondary measure}}
• Uses as position probability density:
  • {{Annotated link|Atomic orbital}}
  • {{Annotated link|Home range}}

References

{{reflist}}

Further reading

• BOOK

• BOOK, George, Casella, George Casella, Roger L., Berger, Roger Lee Berger, Statistical Inference, Second, Thomson Learning, 2002, 0-534-24312-6, 34–37,
• BOOK

External links

• {{Springer

}}

• {{MathWorld|ProbabilityDensityFunction}}

{{Theory of probability distributions}}