zipfextR (version 1.0.2)

zipfpssMean: 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).

Value

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

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

References

Sarabia Alegr<U+00ED>a, J. M., G<U+00F3>mez D<U+00E9>niz, E. M. I. L. I. O., & V<U+00E1>zquez Polo, F. (2007). Estad<U+00ED>stica actuarial: teor<U+00ED>a y aplicaciones. Pearson Prentice Hall.

Examples

# NOT RUN {
zipfpssMean(2.5, 1.3)
zipfpssMean(2.5, 1.3, TRUE)
# }