Statistical Distributions

Linear Distributions

Simple random distributions as the uniform, low-pass, high-pass, and mean are also called linear distributions, because their histograms are linear functions, i.e. we can describe them using linear envelopes:

This graphic shows the histograms (probability curves) of the four linear distributions: red=uniform, green=low-pass, light-blue=high-pass, blue=mean. The X axis represents the range of values for the random selection (0.0, 1.0], the Y axis their probability index (0.0, 1.0). For instance, the value 0.5 has a probability index of 1.0 for the uniform and mean distribution and a probability index of 0.5 for both the high-pass and low-pass distributions.

Non-linear Distributions

These distributions have histograms which are not linear functions as in the previous cases. Most of these non-linear functions use an auxiliary parameter a to control some aspects of the histogram.

Exponential

This distribution is similar to the low-pass but its histogram is exponential. The parameter a works as the 'stretching factor' of the histogram: the larger its value, the steeper the function's curve, and consequently higher is the probability of lower values. The following graphic shows the histograms of this distribution for three values of a.

Color	a
red	2.0
green	1.0
light-blue	0.5

This distribution generates floating point values greater than 0.0 without an upper bound (in theory, it goes up to infinity). The following graphics show the results of the generation of a thousand random values using the exponential distribution. Colors correspond with the ones of the histograms above (for each value of a). The graphics show the range of values [0.0, 1.0) that coincides with histograms range.

We can see that, for this range of values, as the parameter a decreases the distribution of points gets more uniform (see the light-blue graph), and that density of lower values grows with the value of a (see the red graph). We can also see that, as a consequence of filtering values out of the [0.0, 1.0) range, global density decreases with the value of a (compare the density of the three graphics).

Gamma

The shape of the histogram of the gamma distribution is controlled by the a parameter. This parameter can only take integer values greater than one. With a = 1 the gamma histogram looks exactly the same as the exponential one for the same value of a (the green curve in the histograms presented above). As the value of a grows, the histogram takes the shape of an asymmetric bell which peak (or mode, in statistic terms) is close to a-1, and mean close to a. The following graphic shows gamma histograms for three values of a.

Color	a
red	2
green	4
light-blue	8

This distribution, as the exponential, generates floating point values greater than 0.0 without an upper bound (in theory, it goes up to infinity). The following graphic shows the distribution of a thousand numbers for each of the three values of a shown on the histograms (using the same color coding).

We can see that the value of a controls both the location of the density peak as well as its dispersion: with higher values of a the points are spread (see the light-blue dots). In contrast, with lower values of a, the dots tend to be concentrated around the histogram's peak or mode (see the red dots). We can make a parallel between the histogram of this distribution and the frequency response of a resonant filter, where the parameter a controls both the Q and the center frequency.

Poisson

The poisson distribution is different from the rest because its histogram is discrete instead of continuous. This means that it generates positive integer values instead of floating points ones. The parameter a controls in this case the mean value of the distribution. The following graphics show the histogram of the poisson distribution for three values of a.

Color	a
red	2.5
green	4.5
light-blue	6.5

Symmetrical non-linear distributions

The gaussian, cauchy, and beta distributions have symmetrical histograms. The beta distribution is of particular interest because its symmetry can be parametrically altered.

Gaussian

gaussian

normal

Cauchy

cauchy

gaussian

cauchy

Beta

beta

The red curve shows the histogram of the beta distribution for a=b=0.5, the green curve shows the histogram for a=b=0.25.

With different values of a and b the histogram is no longer symmetrical.

The light-blue curve shows the histogram of the beta distribution for a=0.75 and b=0.5, the blue curve shows the histogram for a=0.5 and b=0.75.