Distributacalcul (version 0.2.2)

MGF_beta: Moment Generating Function of the Beta distribution

Description

Moment Generating Function (MGF) of the Beta distribution with shape parameters \(\alpha\) and \(\beta\).

Usage

MGF_beta(t, shape1, shape2, k0)

Arguments

t

t.

shape1

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

shape2

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

k0

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

Value

Function :

• MGF_beta gives the moment generating function (MGF).

• E_beta gives the expected value.

• V_beta gives the variance.

• kthmoment_beta gives the kth moment.

• Etronq_beta gives the truncated mean.

• SL_beta gives the stop-loss.

• Elim_beta gives the limited mean.

• Mexcess_beta gives the mean excess loss.

• TVaR_beta gives the Tail Value-at-Risk.

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

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 \(x \in [0, 1]\), \(\alpha, \beta > 0\).

See Also

Other Beta Distribution: E_beta(), Elim_beta(), Etronq_beta(), Mexcess_beta(), SL_beta(), TVaR_beta(), V_beta(), VaR_beta(), kthmoment_beta()

Examples

# NOT RUN {
MGF_beta(t = 1, shape1 = 3, shape2 = 5, k0 = 1E2)

# }