# Pareto: Pareto Distribution In Distributacalcul: Probability Distribution Functions

## Description

Pareto distribution with shape parameter alpha and rate parameter lambda.

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21``` ```dPareto(x, shape, rate = 1/scale, scale = 1/rate) pPareto(q, shape, rate = 1/scale, scale = 1/rate, lower.tail = TRUE) expValPareto(shape, rate = 1/scale, scale = 1/rate) varPareto(shape, rate = 1/scale, scale = 1/rate) kthMomentPareto(k, shape, rate = 1/scale, scale = 1/rate) expValLimPareto(d, shape, rate = 1/scale, scale = 1/rate) expValTruncPareto(d, shape, rate = 1/scale, scale = 1/rate, less.than.d = TRUE) stopLossPareto(d, shape, rate = 1/scale, scale = 1/rate) meanExcessPareto(d, shape, rate = 1/scale, scale = 1/rate) VatRPareto(kap, shape, rate = 1/scale, scale = 1/rate) TVatRPareto(kap, shape, rate = 1/scale, scale = 1/rate) ```

## Arguments

 `x` vector of quantiles. `shape` shape parameter alpha, must be positive. `rate` rate parameter lambda, must be positive. `scale` alternative parameterization to the rate parameter, scale = 1 / rate. `q` vector of quantiles. `lower.tail` logical; if TRUE (default), probabilities are P[X ≤ x], otherwise, P[X > x]. `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.

## Details

The Pareto distribution with rate parameter lam as well as shape parameter a has density:

f(x) = (a lam^a)/ (lam + x)^(a + 1)

for x > 0, a, lam > 0.

## Value

Function :

• `dPareto` gives the probability density function (PDF).

• `pPareto` gives the cumulative density function (CDF).

• `expValPareto` gives the expected value.

• `varPareto` gives the variance.

• `kthMomentPareto` gives the kth moment.

• `expValLimPareto` gives the limited mean.

• `expValTruncPareto` gives the truncated mean.

• `stopLossPareto` gives the stop-loss.

• `meanExcessPareto` gives the mean excess loss.

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

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

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

## 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 68``` ```# With scale parameter dPareto(x = 2, shape = 2, scale = 5) # With rate parameter dPareto(x = 2, shape = 2, rate = .20) # With scale parameter pPareto(q = 2, shape = 2, scale = 5) # With rate parameter pPareto(q = 2, shape = 2, rate = 0.20) # Survival function pPareto(q = 2, shape = 2, rate = 0.20, lower.tail = FALSE) # With scale parameter expValPareto(shape = 5, scale = 0.5) # With rate parameter expValPareto(shape = 5, rate = 2) # With scale parameter varPareto(shape = 5, scale = 0.5) # With rate parameter varPareto(shape = 5, rate = 2) # With scale parameter kthMomentPareto(k = 4, shape = 5, scale = 0.5) # With rate parameter kthMomentPareto(k = 4, shape = 5, rate = 2) # With scale parameter expValLimPareto(d = 4, shape = 5, scale = 0.5) # With rate parameter expValLimPareto(d = 4, shape = 5, rate = 2) # With scale parameter expValTruncPareto(d = 4, shape = 5, scale = 0.5) # With rate parameter expValTruncPareto(d = 4, shape = 5, rate = 2) # With scale parameter stopLossPareto(d = 2, shape = 5, scale = 0.5) # With rate parameter stopLossPareto(d = 2, shape = 5, rate = 2) # With scale parameter meanExcessPareto(d = 6, shape = 5, scale = 0.5) # With rate parameter meanExcessPareto(d = 6, shape = 5, rate = 2) # With scale parameter VatRPareto(kap = .99, shape = 5, scale = 0.5) # With rate parameter VatRPareto(kap = .99, shape = 5, rate = 2) # With scale parameter TVatRPareto(kap = .99, shape = 5, scale = 0.5) # With rate parameter TVatRPareto(kap = .99, shape = 5, rate = 2) ```

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