VGAM (version 1.1-12)

Zeta: The Zeta Distribution

Description

Density, distribution function, quantile function and random generation for the zeta distribution.

Usage

dzeta(x, shape, log = FALSE)
pzeta(q, shape, lower.tail = TRUE)
qzeta(p, shape)
rzeta(n, shape)

Value

dzeta gives the density,

pzeta gives the distribution function,

qzeta gives the quantile function, and

rzeta generates random deviates.

Arguments

x, q, p, n

Same as Poisson.

shape

The positive shape parameter $s$.

lower.tail, log

Same meaning as in Normal.

Author

T. W. Yee

Details

The density function of the zeta distribution is given by $$y^{-s-1}/\zeta (s+1)$$ where $s>0$, $y=1,2,\dots$, and $\zeta$ is Riemann's zeta function.

References

Johnson N. L., Kotz S., and Balakrishnan N. (1993). Univariate Discrete Distributions, 2nd ed. New York: Wiley.

See Also

zeta, zetaff, Oazeta, Oizeta, Otzeta.

Examples

dzeta(1:20, shape = 2)
myshape <- 0.5
max(abs(pzeta(1:200, myshape) -
    cumsum(1/(1:200)^(1+myshape)) / zeta(myshape+1)))  # Should be 0

if (FALSE)  plot(1:6, dzeta(1:6, 2), type = "h", las = 1,
               col = "orange", ylab = "Probability",
 main = "zeta probability function; orange: shape = 2; blue: shape = 1")
points(0.10 + 1:6, dzeta(1:6, 1), type = "h", col = "blue")