zipfpssVariance: Variance of the Zipf-PSS distribution.

In zipfextR: Zipf Extended Distributions

Description

Computes the variance of the Zipf-PSS distribution for given values of parameters α and λ.

Usage

zipfpssVariance(alpha, lambda, isTruncated = FALSE)

Arguments

alpha	Value of the α parameter (α > 3).
lambda	Value of the λ parameter (λ > 0).
isTruncated	Logical; if TRUE Use the zero-truncated version of the distribution to calculate the expected value (default = FALSE).

Details

The variance of the Zipf-PSS distribution only exists for α values strictly greater than 3. The value is obtained from the law of total variance that says that:

Var[Y] = E[N]\, Var[X] + E[X]^2 \, Var[N],

where X follows a Zipf distribution with parameter α, and N follows a Poisson distribution with parameter λ. From where one has that:

Var[Y] = λ\, \frac{ζ(α - 2)}{ζ(α)}

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

Var[Y^{ZT}] = \frac{λ\, ζ(α)\, ζ(α - 2)\, (1 - e^{-λ}) - λ^2 \, ζ(α - 1)^2 \, e^{-λ}}{ζ(α)^2 \, (1 - e^{-λ})^2}

Value

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

References

Sarabia Alegría, JM. and Gómez Déniz, E. and Vázquez Polo, F. Estadística actuarial: teoría y aplicaciones. Pearson Prentice Hall.

Examples

zipfpssVariance(4.5, 2.3)
zipfpssVariance(4.5, 2.3, TRUE)

zipfextR documentation built on July 8, 2020, 6:23 p.m.