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,
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
Variance and Standard Deviation :
The Variance of the Riemann Zeta distribution can be found using the Variance Formula as follows.
Using the property
Standard Deviation is given by as follows.