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    Ratio Distribution

    Statistical properties

    mean | piecewise | m/(m - 2) | m>2 indeterminate | (otherwise) mode | (m n - 2 n)/(m (n + 2)) standard deviation | piecewise | (sqrt(2) m sqrt(m + n - 2))/(sqrt(m - 4) (m - 2) sqrt(n)) | m>4 indeterminate | (otherwise) variance | piecewise | (2 m^2 (m + n - 2))/((m - 4) (m - 2)^2 n) | m>4 indeterminate | (otherwise) skewness | piecewise | (2 sqrt(2) sqrt(m - 4) (m + 2 n - 2))/((m - 6) sqrt(n) sqrt(m + n - 2)) | m>6 indeterminate | (otherwise)

    Probability density function (PDF)

    piecewise | (m^(m/2) n^(n/2) x^(n/2 - 1) (m + n x)^(1/2 (-m - n)))/beta(n/2, m/2) | x>0 0 | (otherwise)

    Plots of PDF for typical parameters

    Plots of PDF for typical parameters

    Cumulative distribution function (CDF)

    P (X<=x) = piecewise | I_((n x)/(m + n x))(n/2, m/2) | x>0 0 | (otherwise)

    Plots of CDF for typical parameters

    Plots of CDF for typical parameters

    Percentiles

    10th | (m (1/(I_(1, -0.1)^(-1)(0.5 m, 0.5 n)) - 1))/n 25th | (m (1/(I_(1, -0.25)^(-1)(0.5 m, 0.5 n)) - 1))/n 50th | (m (1/(I_(1, -0.5)^(-1)(0.5 m, 0.5 n)) - 1))/n 75th | (m (1/(I_(1, -0.75)^(-1)(0.5 m, 0.5 n)) - 1))/n 90th | (m (1/(I_(1, -0.9)^(-1)(0.5 m, 0.5 n)) - 1))/n

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