code: def _geom_solve_p_from_mu(mu, b):
      
  
      def p_eq(x, mu, b):
          x, mu, b = Decimal(x), Decimal(mu), Decimal(b)
          return ( (x / (1 - x)) - ((b + 1) / (x**-b - 1)) - mu )
  
      # x here is the param raised to the k_agg power, or 1 - p
      return 1 - optim.brentq(p_eq, 1e-16, 100, args=(mu, b), disp=True)
docs: For the geom_uptrunc, given mu and b, return p.
      Ref: Harte 2011, Oxford U Press. Eq. 7.50.