Computes the variance of the Zipf-PSS distribution for given values of parameters \(\alpha\) and \(\lambda\).
zipfpssVariance(alpha, lambda, isTruncated = FALSE)Value of the \(\alpha\) parameter (\(\alpha > 3\)).
Value of the \(\lambda\) parameter (\(\lambda > 0\)).
Logical; if TRUE Use the zero-truncated version of the distribution to calculate the expected value (default = FALSE).
A positive real value corresponding to the variance of the distribution.
The variance of the Zipf-PSS distribution only exists for \(\alpha\) 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 \(\alpha\), and N follows a Poisson distribution with parameter \(\lambda\). From where one has that:
$$Var[Y] = \lambda\, \frac{\zeta(\alpha - 2)}{\zeta(\alpha)}$$ Particularlly, if one is working with the zero-truncated version of the Zipf-PSS distribution. This values is computed as: $$Var[Y^{ZT}] = \frac{\lambda\, \zeta(\alpha)\, \zeta(\alpha - 2)\, (1 - e^{-\lambda}) - \lambda^2 \, \zeta(\alpha - 1)^2 \, e^{-\lambda}}{\zeta(\alpha)^2 \, (1 - e^{-\lambda})^2}$$
Sarabia Alegr<U+00ED>a, JM. and G<U+00F3>mez D<U+00E9>niz, E. and V<U+00E1>zquez Polo, F. Estad<U+00ED>stica actuarial: teor<U+00ED>a y aplicaciones. Pearson Prentice Hall.
# NOT RUN {
zipfpssVariance(4.5, 2.3)
zipfpssVariance(4.5, 2.3, TRUE)
# }
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