Distributacalcul (version 0.2.2)

p_pareto: Cumulative density function of the Pareto distribution

Description

Cumulative density function of the Pareto distribution with shape parameter \(\alpha\) and rate parameter \(\lambda\).

Usage

p_pareto(q, shape, rate = 1/scale, scale = 1/rate, lower.tail = TRUE)

Arguments

q

quantile.

shape

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

rate

\(\lambda\) rate parameter, must be positive.

scale

alternative parameterization to the rate parameter, scale = 1 / rate.

lower.tail

logical; if \codeTRUE (default) probabilities are Pr(M <= k), otherwise, Pr(M > k).

Value

Function :

• d_pareto gives the density function.

• p_pareto gives the cumulative density function.

• E_pareto gives the expected value.

• V_pareto gives the variance.

• kthmoment_pareto gives the kth moment.

• Etronq_pareto gives the truncated mean.

• SL_pareto gives the stop-loss.

• Elim_pareto gives the limited mean.

• Mexcess_pareto gives the mean excess loss.

• TVaR_pareto gives the Tail Value-at-Risk.

• VaR_pareto gives the Value-at-Risk.

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

Details

The Pareto distribution with rate parameter \(\lambda\) as well as shape parameter \(\alpha\) has density: $$f\left(x\right) = \frac{\alpha% \lambda^{\alpha}}{(\lambda + x)^{\alpha + 1}}$$ for \(x \in \mathcal{R}^+\), \(\alpha, \lambda > 0\).

See Also

Other Pareto Distribution: E_pareto(), Elim_pareto(), Etronq_pareto(), Mexcess_pareto(), SL_pareto(), TVaR_pareto(), V_pareto(), VaR_pareto(), d_pareto(), kthmoment_pareto()

Examples

# NOT RUN {
# With scale parameter
p_pareto(q = 2, shape = 2, scale = 5)

# With rate parameter
p_pareto(q = 2, shape = 2, rate = 5)

# Survival function
p_pareto(q = 2, shape = 2, rate = 5, lower.tail = FALSE)

# }