Snapshot 1: beta distribution function with a fixed mode and left skewness

Snapshot 2: symmetric beta distribution function where the mode, mean and median coincide

Snapshot 3: wide beta distribution function with a fixed mean and right skewness

Snapshot 4: narrow beta distribution function with a fixed median and left skewness

Snapshot 5: beta distribution function with a fixed mode and right skewness resembling a truncated distribution, reminiscent of the coarse fraction after sieving the fines

Snapshot 6: beta distribution function with a fixed mean and left skewness resembling a truncated distribution, reminiscent of the fine fraction after sieving

The beta distribution is a convenient flexible function for a random variable

in a finite absolute range from

to

, determined by empirical or theoretical considerations. A corresponding normalized dimensionless independent variable

can be defined by

,

or, when the

spread is over orders of magnitude,

,

which restricts its domain to

in either case.

The beta distribution function, with two parameters

and

, can be written in the form [1–3]

When both

,

is a unimodal distribution. When

it is symmetric around

and for

its skewness direction is determined by whether

or

.

Since for

the beta distribution's mode is

,

,

can be expressed explicitly in terms of

and the mode, mean or median, that is,

respectively. These terms, in turn, can be used to calculate and plot the beta distribution function for any chosen (fixed) value of the mode, mean or median, as a function of

alone.

Choose the parameter to be fixed (mode, mean or median) with the "plot parameter" setter bar and enter its selected value using the slider. The value of

can then also be entered and varied with its slider to calculate and plot the PDF and CDF forms of the beta distribution for the current setting. The numerical values of the other parameters and corresponding variance are then calculated and displayed above the plots.

Except for the mode, the mean, median and variance can also be calculated with the built-in Wolfram Language functions

Mean,

Median and

Variance, and for simplicity we have used the median's commonly accepted approximation formula for

and not the more elaborate general form [4]. Also note that certain entered control settings may produce parameters that violate the condition

and thus should be discarded.