a) Cauchy distribution:
b) Power law:
We have already defined discrete power law distributions. We present here a continuous analog. Our starting point is to introduce tail probabilities that decay according to power law:
FX(x) = 0, otherwise. In order for X to be a continuous random variable, FX cannot have a jump at x = c, and we therefore need β = cα. The corresponding density is of the form
EXPECTED VALUES:
Similar to the discrete case, given a continuous random variable X with PDF fX, we define