erlang.Rd
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)
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 |
kap | probability. |
t | t. |
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.
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 \(x \in \mathcal{R}^+\), \(\beta > 0\), \(n \in \mathcal{N}^+\).
Function VatRErlang is a wrapper of the qgamma
function from the stats package.
dErlang(x = 2, shape = 2, scale = 4)#> [1] 0.07581633pErlang(q = 2, shape = 2, scale = 4)#> [1] 0.09020401expValErlang(shape = 2, scale = 4)#> [1] 8varErlang(shape = 2, scale = 4)#> [1] 32kthMomentErlang(k = 3, shape = 2, scale = 4)#> [1] 1536expValLimErlang(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] 7.884899stopLossErlang(d = 2, shape = 2, scale = 4)#> [1] 6.065307meanExcessErlang(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] 13.94824mgfErlang(t = 2, shape = 2, scale = .25)#> [1] 4