A note on various holding probabilities for random lazy random walks on finite groups
For \(a, b > 0\), let \(F(x)=\int^x_0 y^{a-1}(1 - y)^{b-1}dy/B(a,b)\), \(0\leq x\leq 1\), denote the beta distribution function, and set \(G(y) = F^{-1}(y)\), \(0\leq y\leq 1\). The author calls \(G\) the complementary beta distribution function. He derives some attractive properties of \(G\), most ones being complementary to those of \(F\). For example, let \(X_{i:n}\) denote the \(i\)th order statistic from an i.i.d. sample of size \(n\) from the distribution function \(G\). Then the expectation of the spacing \(S_{i:n}=X_{i+1:n}- X_{i:n}\) is \(E(S_{i:n}) = {n\choose i}B(a+i, b+n-i)/B(a,b)\). Good measures of the skewness and kurtosis of \(G\) are also provided.
complementary beta distribution function
scientific article; zbMATH DE number 1796679