Normal Distribution

Normal distribution

expValNorm(mean, sd)

varNorm(mean, sd)

expValLimNorm(d, mean = 0, sd = 1)

expValTruncNorm(d, mean = 0, sd = 1, less.than.d = TRUE)

stopLossNorm(d, mean = 0, sd = 1)

meanExcessNorm(d, mean = 0, sd = 1)

VatRNorm(kap, mean = 0, sd = 1)

TVatRNorm(kap, mean = 0, sd = 1)

mgfNorm(t, mean = 0, sd = 1)

Arguments

mean

mean (location) parameter \(\mu\).
sd

standard deviation \(\sigma\), must be positive.
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 :

  • expValNorm gives the expected value.

  • varNorm gives the variance.

  • expValLimNorm gives the limited mean.

  • expValTruncNorm gives the truncated mean.

  • stopLossNorm gives the stop-loss.

  • meanExcessNorm gives the mean excess loss.

  • VatRNorm gives the Value-at-Risk.

  • TVatRNorm gives the Tail Value-at-Risk.

  • mgfNorm gives the moment generating function (MGF).

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

Details

The Normal distribution with mean \(\mu\) and standard deviation \(\sigma\) has density: $$\frac{1}{\sqrt{2\pi}\sigma}\textrm{e}^{-\frac{1}{2}\left(\frac{x - \mu}{\sigma}\right)^2}$$ for \(x \in \mathcal{R}\), \(\mu \in \mathcal{R}, \sigma > 0\).

Note

Function VatRNorm is a wrapper of the qnorm function from the stats package.

Examples

expValNorm(mean = 3, sd = 5)
#> [1] 3

varNorm(mean = 3, sd = 5)
#> [1] 25

expValLimNorm(d = 2, mean = 2, sd = 5)
#> [1] 0.005288598

expValTruncNorm(d = 2, mean = 2, sd = 5)
#> [1] -0.9947114

stopLossNorm(d = 2, mean = 2, sd = 5)
#> [1] 0.005288598

meanExcessNorm(d = 2, mean = 2, sd = 5)
#> [1] 0.0105772

VatRNorm(kap = 0.8, mean = 3, sd = 5)
#> [1] 7.208106

TVatRNorm(kap = 0.8, mean = 2, sd = 5)
#> [1] 8.999048

mgfNorm(t = 1, mean = 3, sd = 5)
#> [1] 5389698