Best fitting duty cycle values f_duty(z_i) (upper panel) and power law indices Γ(z_i) from Eq. (1) (lower panel) in nine redshift bins, shown with red crosses. In the upper panel the power law index is assumed to be in parametric form Γ(z) = c₄ + c₅z while the amplitude factors in each redshift bins are assumed to be free parameters, i.e., including M₀ the fitting problem has 12 free parameters in total. With solid black line we show the best fitting three parameter analytic function with the detailed form as specified in the legend. In the lower panel f_duty(z) is fixed to the above analytic form while the power law indices in each redshift bin Γ(z_i) are allowed to vary freely, i.e., this time we have 13 free parameters. Solid black and gray lines show linear and quadratic polynomial approximations to best fitting Γ(z_i). The dotted lines and gray shaded regions represent best fit values along with 1σ intervals in case all Γ(z_i) and f_duty(z_i) are all allowed to be free (19 free parameters).