Distribution Function

The distribution function $D(x)$, sometimes also called the Probability Distribution Function, describes the probability that a trial $X$ takes on a value less than or equal to a number $x$. The distribution function is therefore related to a continuous Probability Density Function $P(x)$ by

$$D(x) = P(X\leq x) \equiv \int^x_{-\infty} P(x')\,dx',$$ (1)

so $P(x)$ (when it exists) is simply the derivative of the distribution function

$$P(x)=D'(x) = [P(x')]_{-\infty}^x = P(x)-P(-\infty).$$ (2)

Similarly, the distribution function is related to a discrete probability $P(x)$ by

$$D(x) = P(X \leq x) = \sum_{X \leq x} P(x).$$ (3)

In general, there exist distributions which are neither continuous nor discrete.

A Joint Distribution Function can be defined if outcomes are dependent on two parameters:

$$D(x,y) \equiv P(X\leq x, Y\leq y)$$ (4)

$$D_x(x) \equiv D(x, \infty)$$ (5)

$$D_y(y) \equiv D(\infty, y).$$ (6)

Similarly, a multiple distribution function can be defined if outcomes depend on $n$ parameters:

$$D(a_1, \ldots, a_n) \equiv P(x_1\leq a_1, \ldots, x_n\leq a_n).$$ (7)

Given a continuous $P(x)$, assume you wish to generate numbers distributed as $P(x)$ using a random number generator. If the random number generator yields a uniformly distributed value $y_i$ in [0,1] for each trial $i$, then compute

$$D(x)\equiv \int^x P(x')\,dx'.$$ (8)

The Formula connecting $y_i$ with a variable distributed as $P(x)$ is then

$$x_i=D^{-1}(y_i),$$ (9)

where $D^{-1}(x)$ is the inverse function of $D(x)$. For example, if $P(x)$ were a Gaussian Distribution so that

$$D(x)={1\over 2}\left[{1+{\rm erf}\left({x-\mu\over \sigma\sqrt{2}}\right)}\right],$$ (10)

then

$$x_i=\sigma\sqrt{2}\,{\rm erf}\,{}^{-1}(2y_i-1)+\mu.$$ (11)

If $P(x)=Cx^n$ for $x\in (x_{\rm min},x_{\rm max})$, then normalization gives

$$\int_{x_{\rm min}}^{x_{\rm max}} P(x)\,dx=C{[x^{n+1}]^{x_{\rm max}}_{x_{\rm min}}\over n+1}=1,$$ (12)

so

$$C={n+1\over{x_{\rm max}}^{n+1}-{x_{\rm min}}^{n+1}}.$$ (13)

Let $y$ be a uniformly distributed variate on $[0,1]$. Then

$$D(x) = \int_{x_{\rm min}}^x P(x)\,dx = C\int_{x_{\rm min}}^x x^n\,dx$$

$$= {C\over n+1}(x^{n+1}-{x_{\rm min}}^{n+1})\equiv y,$$ (14)

and the variate given by

$$x = \left({{n+1\over C} y+{x_{\rm min}}^{n+1}}\right)^{1/(n+1)}$$

$$= [({x_{\rm max}}^{n+1}-{x_{\rm min}}^{n+1})y+{x_{\rm min}}^{n+1}]^{1/(n+1)}$$ (15)

is distributed as $P(x)$.

A distribution with constant Variance of $y$ for all values of $x$ is known as a Homoscedastic distribution. The method of finding the value at which the distribution is a maximum is known as the Maximum Likelihood method.

See also Bernoulli Distribution, Beta Distribution, Binomial Distribution, Bivariate Distribution, Cauchy Distribution, Chi Distribution, Chi-Squared Distribution, Cornish-Fisher Asymptotic Expansion, Correlation Coefficient, Distribution, Double Exponential Distribution, Equally Likely Outcomes Distribution, Exponential Distribution, Extreme Value Distribution, F-Distribution, Fermi-Dirac Distribution, Fisher's z-Distribution, Fisher-Tippett Distribution, Gamma Distribution, Gaussian Distribution, Geometric Distribution, Half-Normal Distribution, Hypergeometric Distribution, Joint Distribution Function, Laplace Distribution, Lattice Distribution, Lévy Distribution, Logarithmic Distribution, Log-Series Distribution, Logistic Distribution, Lorentzian Distribution, Maxwell Distribution, Negative Binomial Distribution, Normal Distribution, Pareto Distribution, Pascal Distribution, Pearson Type III Distribution, Poisson Distribution, Pólya Distribution, Ratio Distribution, Rayleigh Distribution, Rice Distribution, Snedecor's F-Distribution, Student's t-Distribution, Student's z-Distribution, Uniform Distribution, Weibull Distribution

References

Abramowitz, M. and Stegun, C. A. (Eds.). ``Probability Functions.'' Ch. 26 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 925-964, 1972.

Iyanaga, S. and Kawada, Y. (Eds.). ``Distribution of Typical Random Variables.'' Appendix A, Table 22 in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, pp. 1483-1486, 1980.