There are large cuts at surfaces intersecting the lines α = 1, β = 1, α = 2, and β = 2 because at these values the beta distribution change from 2 modes, to 1 mode to no mode. The beta density function can take a wide variety of different shapes depending on the values of the two parameters "α" and "β". The ability of the beta distribution to take this great diversity of shapes (using only two parameters) is partly responsible for finding wide application for modeling actual measurements:   The density function is skewed. An interchange of parameter values yields the mirror image (the reverse) of the initial curve, some more specific cases:       Two unknown parameters (formula_220 of a beta distribution supported in the [0,1] interval) can be estimated, using the method of moments, with the first two moments (sample mean and sample variance) as follows. Let:  be the sample mean estimate and  be the sample variance estimate. The method-of-moments estimates of the parameters are  When the distribution is required over a known interval other than [0, 1] with random variable "X", say ["a", "c"] with random variable "Y", then replace formula_227 with formula_228 and formula_229 with formula_230 in the above couple of equations for the shape parameters (see the "Alternative parametrizations, four parameters" section below)., where:  All four parameters (formula_233 of a beta distribution supported in the ["a", "c"] interval -see section "Alternative parametrizations, Four parameters"-) can be estimated, using the method of moments developed by Karl Pearson, by equating sample and population values of the first four central moments (mean, variance, skewness and excess kurtosis). The excess kurtosis was expressed in terms of the square of the skewness, and the sample size ν = α + β, (see previous section "Kurtosis") as follows:  One can use this equation to solve for the sample size ν= α + β in terms of the square of the skewness and the excess kurtosis as follows:  This is the ratio (multiplied by a factor of 3) between the previously derived limit boundaries for the beta distribution in a space (as originally done by Karl Pearson) defined with coordinates of the square of the skewness in one axis and the excess kurtosis in the other axis (see previous section titled "Kurtosis bounded by the square of the skewness"):  The case of zero skewness, can be immediately solved because for zero skewness, α = β and hence ν = 2α = 2β, therefore α = β = ν/2  (Excess kurtosis is negative for the beta distribution with zero skewness, ranging from -2 to 0, so that formula_239 -and therefore the sample shape parameters- is positive, ranging from zero when the shape parameters approach zero and the excess kurtosis approaches -2, to infinity when the shape parameters approach infinity and the excess kurtosis approaches zero).