# Gamma: Gamma Distribution In Distributacalcul: Probability Distribution Functions

## Description

Gamma distribution with shape parameter alpha and rate parameter beta.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19``` ```expValGamma(shape, rate = 1/scale, scale = 1/rate) varGamma(shape, rate = 1/scale, scale = 1/rate) kthMomentGamma(k, shape, rate = 1/scale, scale = 1/rate) expValLimGamma(d, shape, rate = 1/scale, scale = 1/rate) expValTruncGamma(d, shape, rate = 1/scale, scale = 1/rate, less.than.d = TRUE) stopLossGamma(d, shape, rate = 1/scale, scale = 1/rate) meanExcessGamma(d, shape, rate = 1/scale, scale = 1/rate) VatRGamma(kap, shape, rate = 1/scale, scale = 1/rate) TVatRGamma(kap, shape, rate = 1/scale, scale = 1/rate) mgfGamma(t, shape, rate = 1/scale, scale = 1/rate) ```

## Arguments

 `shape` shape parameter alpha, must be positive. `rate` rate parameter beta, must be positive. `scale` alternative parameterization to the rate parameter, scale = 1 / rate. `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 Gamma distribution with shape parameter a and rate parameter b has density:

f(x) = b^a / Γ(a) x^{a - 1} e^{-b x}

for x > 0, b, a > 0.

## Value

Function :

• `expValGamma` gives the expected value.

• `varGamma` gives the variance.

• `kthMomentGamma` gives the kth moment.

• `expValLimGamma` gives the limited mean.

• `expValTruncGamma` gives the truncated mean.

• `stopLossGamma` gives the stop-loss.

• `meanExcessGamma` gives the mean excess loss.

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

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

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

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

## Note

Function VatRGamma is a wrapper for the `qgamma` function 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 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67``` ```# With scale parameter expValGamma(shape = 3, scale = 4) # With rate parameter expValGamma(shape = 3, rate = 0.25) # With scale parameter varGamma(shape = 3, scale = 4) # With rate parameter varGamma(shape = 3, rate = 0.25) # With scale parameter kthMomentGamma(k = 2, shape = 3, scale = 4) # With rate parameter kthMomentGamma(k = 2, shape = 3, rate = 0.25) # With scale parameter expValLimGamma(d = 2, shape = 3, scale = 4) # With rate parameter expValLimGamma(d = 2, shape = 3, rate = 0.25) # With scale parameter expValTruncGamma(d = 2, shape = 3, scale = 4) # With rate parameter expValTruncGamma(d = 2, shape = 3, rate = 0.25) # values greather than d expValTruncGamma(d = 2, shape = 3, rate = 0.25, less.than.d = FALSE) # With scale parameter stopLossGamma(d = 2, shape = 3, scale = 4) # With rate parameter stopLossGamma(d = 2, shape = 3, rate = 0.25) # With scale parameter meanExcessGamma(d = 2, shape = 3, scale = 4) # With rate parameter meanExcessGamma(d = 2, shape = 3, rate = 0.25) # With scale parameter VatRGamma(kap = .2, shape = 3, scale = 4) # With rate parameter VatRGamma(kap = .2, shape = 3, rate = 0.25) # With scale parameter TVatRGamma(kap = .2, shape = 3, scale = 4) # With rate parameter TVatRGamma(kap = .2, shape = 3, rate = 0.25) mgfGamma(t = 1, shape = 3, rate = 5) ```

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