zipfpssMean {zipfextR}

Expected value of the Zipf-PSS distribution.

Description

Computes the expected value of the Zipf-PSS distribution for given values of parameters \alpha and \lambda.

Usage

zipfpssMean(alpha, lambda, isTruncated = FALSE)

Arguments

alpha

Value of the \alpha parameter (\alpha > 2).

lambda

Value of the \lambda parameter (\lambda > 0).

isTruncated

Logical; if TRUE Use the zero-truncated version of the distribution to calculate the expected value (default = FALSE).

Details

The expected value of the Zipf-PSS distribution only exists for \alpha values strictly greater than 2. The value is obtained from the law of total expectation that says that:

E[Y] = E[N]\, E[X],

where E[X] is the mean value of the Zipf distribution and E[N] is the expected value of a Poisson one. From where one has that:

E[Y] = \lambda\, \frac{\zeta(\alpha - 1)}{\zeta(\alpha)}

Particularlly, if one is working with the zero-truncated version of the Zipf-PSS distribution. This values is computed as:

E[Y^{ZT}] = \frac{\lambda\, \zeta(\alpha - 1)}{\zeta(\alpha)\, (1 - e^{-\lambda})}

Value

A positive real value corresponding to the mean value of the distribution.

References

Sarabia Alegría, J. M., Gómez Déniz, E. M. I. L. I. O., & Vázquez Polo, F. (2007). Estadística actuarial: teoría y aplicaciones. Pearson Prentice Hall.

Examples

zipfpssMean(2.5, 1.3)
zipfpssMean(2.5, 1.3, TRUE)