# erlang: Erlang Distribution In Distributacalcul: Probability Distribution Functions

## Description

Erlang distribution with shape parameter n and rate parameter beta.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23``` ```dErlang(x, shape, rate = 1/scale, scale = 1/rate) pErlang(q, shape, rate = 1/scale, scale = 1/rate, lower.tail = TRUE) expValErlang(shape, rate = 1/scale, scale = 1/rate) varErlang(shape, rate = 1/scale, scale = 1/rate) kthMomentErlang(k, shape, rate = 1/scale, scale = 1/rate) expValLimErlang(d, shape, rate = 1/scale, scale = 1/rate) expValTruncErlang(d, shape, rate = 1/scale, scale = 1/rate, less.than.d = TRUE) stopLossErlang(d, shape, rate = 1/scale, scale = 1/rate) meanExcessErlang(d, shape, rate = 1/scale, scale = 1/rate) VatRErlang(kap, shape, rate = 1/scale, scale = 1/rate) TVatRErlang(kap, shape, rate = 1/scale, scale = 1/rate) mgfErlang(t, shape, rate = 1/scale, scale = 1/rate) ```

## Arguments

 `x` vector of quantiles. `shape` shape parameter n, must be a positive integer. `rate` rate parameter beta, must be positive. `scale` alternative parameterization to the rate parameter, scale = 1 / rate. `q` vector of quantiles. `lower.tail` logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x]. `k` kth-moment. `d` cut-off value. `less.than.d` logical; if `TRUE` (default) truncated mean for values <= d, otherwise, for values > d. `kap` probability. `t` t.

## Details

The Erlang distribution with shape parameter n and rate parameter b has density:

f(x) = b^n / Γ(n) x^{n - 1} e^{-b x}

for x > 0, b > 0, n = 1, 2, 3, ....

## Value

Function :

• `dErlang` gives the probability density function (PDF).

• `pErlang` gives the cumulative density function (CDF).

• `expValErlang` gives the expected value.

• `varErlang` gives the variance.

• `kthMomentErlang` gives the kth moment.

• `expValLimErlang` gives the limited mean.

• `expValTruncErlang` gives the truncated mean.

• `stopLossErlang` gives the stop-loss.

• `meanExcessErlang` gives the mean excess loss.

• `VatRErlang` gives the Value-at-Risk.

• `TVatRErlang` gives the Tail Value-at-Risk.

• `mgfErlang` gives the moment generating function (MGF).

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

## Note

Function VatRErlang is a wrapper of the `qgamma` function from the stats package.

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35``` ```dErlang(x = 2, shape = 2, scale = 4) pErlang(q = 2, shape = 2, scale = 4) expValErlang(shape = 2, scale = 4) varErlang(shape = 2, scale = 4) kthMomentErlang(k = 3, shape = 2, scale = 4) expValLimErlang(d = 2, shape = 2, scale = 4) # With rate parameter expValTruncErlang(d = 2, shape = 2, scale = 4) # Values greater than d expValTruncErlang(d = 2, shape = 2, scale = 4, less.than.d = FALSE) stopLossErlang(d = 2, shape = 2, scale = 4) meanExcessErlang(d = 3, shape = 2, scale = 4) # With scale parameter VatRErlang(kap = .2, shape = 2, scale = 4) # With rate parameter VatRErlang(kap = .2, shape = 2, rate = 0.25) # With scale parameter TVatRErlang(kap = .2, shape = 3, scale = 4) # With rate parameter TVatRErlang(kap = .2, shape = 3, rate = 0.25) mgfErlang(t = 2, shape = 2, scale = .25) ```

Distributacalcul documentation built on Sept. 13, 2020, 5:19 p.m.