Logarithmic Distribution

Logarithmic distribution with probability parameter \(\gamma\).

dLogarithmic(x, prob)

pLogarithmic(q, prob, lower.tail = TRUE)

expValLogarithmic(prob)

varLogarithmic(prob)

VatRLogarithmic(kap, prob)

mgfLogarithmic(t, prob)

pgfLogarithmic(t, prob)

Arguments

x

vector of quantiles.
prob

probability parameter \(\gamma\).
q

vector of quantiles.
lower.tail

logical; if TRUE (default), probabilities are \(P[X \le x]\), otherwise, \(P[X > x]\).
kap

probability.
t

t.

Value

Function :

  • dLogarithmic gives the probability density function (PDF).

  • pLogarithmic gives the cumulative density function (CDF).

  • expValLogarithmic gives the expected value.

  • varLogarithmic gives the variance.

  • VatRLogarithmic gives the Value-at-Risk.

  • mgfLogarithmic gives the moment generating function (MGF).

  • pgfLogarithmic gives the probability generating function (MGF).

Invalid parameter values will return an error detailing which parameter is problematic.

Details

The Logarithmic distribution with probability parameter \(\gamma\) has probability mass function : $$Pr(X = k) = \frac{-\gamma^{k}}{\ln(1 - \gamma)k}$$, for \(k = 0, 1, 2, \dots\), and \(\gamma \in (0, 1)\)].

Examples

dLogarithmic(x = 3, prob = 0.2)
#> [1] 0.01195045

pLogarithmic(q = 3, prob = 0.2)
#> [1] 0.9978629

expValLogarithmic(prob = 0.50)
#> [1] 1.442695

varLogarithmic(prob = 0.50)
#> [1] -1.608042

VatRLogarithmic(kap = 0.99, prob = 0.2)
#> [1] 3

mgfLogarithmic(t = .2, prob = 0.50)
#> [1] 1.361051

pgfLogarithmic(t = .2, prob = 0.50)
#> [1] 0.1520031