Distributacalcul (version 0.4.0)

Beta: Beta Distribution

Description

Beta distribution with shape parameters \(\alpha\) and \(\beta\).

Usage

expValBeta(shape1, shape2)
varBeta(shape1, shape2)
kthMomentBeta(k, shape1, shape2)
expValLimBeta(d, shape1, shape2)
expValTruncBeta(d, shape1, shape2, less.than.d = TRUE)
stopLossBeta(d, shape1, shape2)
meanExcessBeta(d, shape1, shape2)
VatRBeta(kap, shape1, shape2)
TVatRBeta(kap, shape1, shape2)
mgfBeta(t, shape1, shape2, k0)

Value

Function :

• expValBeta gives the expected value.

• varBeta gives the variance.

• kthMomentBeta gives the kth moment.

• expValLimBeta gives the limited mean.

• expValTruncBeta gives the truncated mean.

• stopLossBeta gives the stop-loss.

• meanExcessBeta gives the mean excess loss.

• VatRBeta gives the Value-at-Risk.

• TVatRBeta gives the Tail Value-at-Risk.

• mgfBeta gives the moment generating function (MGF).

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

Arguments

shape1

shape parameter \(\alpha\), must be positive.

shape2

shape parameter \(\beta\), must be positive.

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.

k0

point up to which to sum the distribution for the approximation.

Details

The Beta distribution with shape parameters \(\alpha\) and \(\beta\) has density: $$f\left(x\right) = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha) % \Gamma(\beta)} x^{\alpha - 1} (1 - x)^(\beta - 1)$$ for \(x \in [0, 1]\), \(\alpha, \beta > 0\).

Examples

expValBeta(shape1 = 3, shape2 = 5)

varBeta(shape1 = 4, shape2 = 5)

kthMomentBeta(k = 3, shape1 = 4, shape2 = 5)

expValLimBeta(d = 0.3, shape1 = 4, shape2 = 5)

expValTruncBeta(d = 0.4, shape1 = 4, shape2 = 5)

# Values less than d
expValTruncBeta(d = 0.4, shape1 = 4, shape2 = 5, less.than.d = FALSE)

stopLossBeta(d = 0.3, shape1 = 4, shape2 = 5)

meanExcessBeta(d = .3, shape1 = 4, shape2 = 5)

VatRBeta(kap = .99, shape1 = 4, shape2 = 5)

TVatRBeta(kap = .99, shape1 = 4, shape2 = 5)

mgfBeta(t = 1, shape1 = 3, shape2 = 5, k0 = 1E2)