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 \(s>0\), \(y=1,2,\ldots\), 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")