The Beta law on \\( {[0,1]} \\) with parameters \\( {a>0} \\) and \\( {b>0} \\) has density<\/p>\n
\\[ x\\mapsto \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)} x^{a-1}(1-x)^{b-1}\\mathbf{1}_{[0,1]}(x). \\]<\/p>\n
It is sometimes more convenient to work on the interval \\( {[-1,1]} \\) instead of \\( {[0,1]} \\). Indeed, with the substitution \\( {y=2x-1} \\), we get the Beta density on \\( {[-1,1]} \\) :<\/p>\n
\\[ y\\mapsto \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)}2^{1-a-b} (1+y)^{a-1}(1-y)^{b-1}\\mathbf{1}_{[-1,1]}(y). \\]<\/p>\n
In particular, when \\( {a=b} \\) then it boils down to<\/p>\n
\\[ y\\mapsto \\frac{\\Gamma(2a)}{\\Gamma(a)^2}2^{1-2a} (1-y^2)^{a-1}\\mathbf{1}_{[-1,1]}(y). \\]<\/p>\n
For \\( {a=1\/2} \\) we get the arcsine<\/b> law, for \\( {a=1} \\) the uniform<\/b> law, and for \\( {a=3\/2} \\) the semicircle<\/b> law. The Jacobi polynomials<\/b> are orthogonal for this \\( {(a,b)} \\)-beta law on \\( {[-1,1]} \\). We recover the Chebyshev polynomials<\/b> of the first and second kind in the arcsine and semicircle cases respectively, and the Legendre polynomials<\/b> in the uniform case.<\/p>\n","protected":false},"excerpt":{"rendered":"
The Beta law on \\( {[0,1]} \\) with parameters \\( {a>0} \\) and \\( {b>0} \\) has density \\[ x\\mapsto \\frac{\\Gamma(a+b)}{\\Gamma(a)\\Gamma(b)} x^{a-1}(1-x)^{b-1}\\mathbf{1}_{[0,1]}(x). \\] It is…<\/p>\n