Erlang Distribution

Erlang distribution with shape parameter \(n\) and rate parameter \(\beta\).

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 \(P[X \le x]\), 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.

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.

Details

The Erlang distribution with shape parameter \(n\) and rate parameter \(\beta\) has density: $$f\left(x\right) = \frac{\beta^{n}}{\Gamma(n)} x^{n - 1}% \mathrm{e}^{-\beta x}$$ for \(x \in \mathcal{R}^+\), \(\beta > 0\), \(n \in \mathcal{N}^+\).

Note

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

Examples

dErlang(x = 2, shape = 2, scale = 4)
#> [1] 0.07581633

pErlang(q = 2, shape = 2, scale = 4)
#> [1] 0.09020401

expValErlang(shape = 2, scale = 4)
#> [1] 8

varErlang(shape = 2, scale = 4)
#> [1] 32

kthMomentErlang(k = 3, shape = 2, scale = 4)
#> [1] 1536

expValLimErlang(d = 2, shape = 2, scale = 4)
#> [1] 1.934693

# With rate parameter
expValTruncErlang(d = 2, shape = 2, scale = 4)
#> [1] 0.1151014

# Values greater than d
expValTruncErlang(d = 2, shape = 2, scale = 4, less.than.d = FALSE)
#> [1] 7.884899

stopLossErlang(d = 2, shape = 2, scale = 4)
#> [1] 6.065307

meanExcessErlang(d = 3, shape = 2, scale = 4)
#> [1] 6.285714

# With scale parameter
VatRErlang(kap = .2, shape = 2, scale = 4)
#> [1] 3.297553

# With rate parameter
VatRErlang(kap = .2, shape = 2, rate = 0.25)
#> [1] 3.297553

# With scale parameter
TVatRErlang(kap = .2, shape = 3, scale = 4)
#> [1] 13.94824

# With rate parameter
TVatRErlang(kap = .2, shape = 3, rate = 0.25)
#> [1] 13.94824

mgfErlang(t = 2, shape = 2, scale = .25)
#> [1] 4