Distributacalcul (version 0.4.0)

erlang: Erlang Distribution

Description

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

Usage

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)

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.

Arguments

x, q

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.

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.

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 \(x \in \mathcal{R}^+\), \(\beta > 0\), \(n \in \mathcal{N}^+\).

Examples

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)