Mathematics | Reimann Zeta Distribution Model

Introduction :

Suppose an event can occur several times within a given unit of time. When the total number of occurrences of the event is unknown, we can think of it as a random variable. When a random variable X takes on values on discrete time interval from 1 to infinity, one choice of a probability density is the Reimann Zeta distribution whose probability density function is given by as follows.

Above expression will be applicable only when given below condition will follow.

 x = 1,2,3,..... 
 f(x) = 0, Otherwise

Where,  is the parameter and  is the value of the zeta function, defined by as follows.

The random variable X following Reimann Zeta Distribution is represented as follows.

X ~ RIE()

Expected Value :

The Expected Value of the Reimann Zeta distribution can be found by summing up products of Values with their respective probabilities as follows.

Using the property , we get the following expression as follows.

Variance and Standard Deviation : 

The Variance of the Riemann Zeta distribution can be found using the Variance Formula as follows.

Using the property , we get the following expression as follows.

Standard Deviation is given by as follows.