# Distributacalcul

The goal of Distributacalcul is to simplify the life of students and scientists by offering premade functions for various functions of probability distributions. Functions calculate moments of the distribution (mean, variance, kth moment) as well as the expected value of functions of the distribution (truncated mean, stop-loss, mean excess loss, etc.). In addition, the package includes some risk measures (Value-at-Risk and Tail Value-at-Risk).

In addition, this package has recently added probability functions for various bivariate copulas. Functions calculate the density associated with the copula, the distribution function and also simulations.

This package depends primarily the stats package.

## Installation

You can install the released version of Distributacalcul from CRAN with:

``````install.packages("Distributacalcul")
``````

And the development version from GitHub with:

``````# install.packages("devtools")
devtools::install_github("alec42/Distributacalcul_Package")
``````

## Example

This is a basic example which shows you the typical use of the functions:

``````library(Distributacalcul)
##  Various moments:
expValBeta(shape1 = 2, shape2 = 4) # E[X]
#> [1] 0.3333333
varBeta(shape1 = 2, shape2 = 4) # V(X)
#> [1] 0.03174603
kthMomentBeta(k = 3, shape1 = 2, shape2 = 4) # E[X^k]
#> [1] 0.07142857

##  Expected value of functions:
expValLimBeta(d = 0.3, shape1 = 2, shape2 = 4) # E[min(X; d)]
#> [1] 2.19811
expValTruncBeta(d = .3, shape1 = 2, shape2 = 4, less.than.d = TRUE) # E[X * 1_{X <= d}]
#> [1] 0.08523
expValTruncBeta(d = .3, shape1 = 2, shape2 = 4, less.than.d = FALSE) # E[X * 1_{X > d}]
#> [1] 0.2481033
meanExcessBeta(d = .3, shape1 = 2, shape2 = 4) # E[(X - d | X > d)]
#> [1] 0.169697
stopLossBeta(d = .3, shape1 = 2, shape2 = 4) # E[max(X - d, 0)]
#> [1] 0.4065693

##  Risk measures:
TVatRBeta(kap = 0.99, shape1 = 2, shape2 = 4) # TVaR_{k}(X)
#> [1] 0.8239414
VatRBeta(kap = 0.99, shape1 = 2, shape2 = 4) # VaR_{k}(X) = F_X^(-1)(k)
#> [1] 0.7779277
``````

## Syntax:

| Function | Syntax | |---------------------------------|-------------------------| | Mean | expValDistribution | | K-th moment | kthMomentDistribution | | Truncated mean | expValTruncDistribution | | Limited expected value | expValLimDistribution | | Variance | varDistribution | | Stop-loss | stopLossDistribution | | Excess of mean | meanExcessDistribution | | Moment Generating Function | mgfDistribution | | Probability Generating Function | pgfDistribution | | Density | dDistribution | | Cumulative density function | pDistribution | | Value-at-Risk (percentile) | VatRDistribution | | Tail Value-at-Risk | TVatRDistribution | | Copula Density | cdCopula | | Copula Distribution Function | cCopula | | Copula Simulation Function | crCopula |

# Included distributions and functions

## Continuous distributions

|   | Erlang | Inverse Gaussian | Weibull | Burr | |:-------------------------------------------:|:------:|:----------------:|:-------:|:----:| | Mean | X | X | X | X | | kth moment | X | | X | X | | Variance | X | X | X | X | | Truncated mean | X | X | X | X | | Limited mean | X | X | X | X | | Stop-loss | X | X | X | X | | Excess of mean | X | X | X | X | | Moment Generating Function | X | X | | | | Probability Density Function | X | | | | | Cumulative Probability Density Function | X | | | | | Value-at-Risk | X | X | X | X | | Tail Value-at-Risk | X | X | X | X |

Table continues below

|   | Log-logistic | Beta | Gamma | Pareto | |:-------------------------------------------:|:------------:|:----:|:-----:|:------:| | Mean | X | X | X | X | | kth moment | X | X | X | X | | Variance | X | X | X | X | | Truncated mean | X | X | X | X | | Limited mean | X | X | X | X | | Stop-loss | X | X | X | X | | Excess of mean | X | X | X | X | | Moment Generating Function | | X | X | | | Probability Density Function | X | | | X | | Cumulative Probability Density Function | X | | | X | | Value-at-Risk | X | X | X | X | | Tail Value-at-Risk | X | X | X | X |

Table continues below

|   | Lognormal | Exponential | Uniform | Normal | |:-------------------------------------------:|:---------:|:-----------:|:-------:|:------:| | Mean | X | X | X | X | | kth moment | X | X | X | | | Variance | X | X | X | X | | Truncated mean | X | X | X | X | | Limited mean | X | X | X | X | | Stop-loss | X | X | X | X | | Excess of mean | X | X | X | X | | Moment Generating Function | | X | X | X | | Probability Density Function | | | | | | Cumulative Probability Density Function | | | | | | Value-at-Risk | X | X | X | X | | Tail Value-at-Risk | X | X | X | X |

## Discrete distributions

|   | Binomial | Negative Binomial | Poisson | |:-------------------------------------------:|:--------:|:-----------------:|:-------:| | Mean | X | X | X | | kth moment | | | | | Variance | X | X | X | | Truncated mean | X | X | X | | Limited mean | | | | | Stop-loss | | | | | Excess of mean | | | | | Moment Generating Function | X | X | X | | Probability Generating Function | X | X | X | | Probability Density Function | | | | | Cumulative Probability Density Function | | | | | Value-at-Risk | X | | | | Tail Value-at-Risk | X | X | X |

Table continues below

|   | Uniform | Logarithmic | Hypergeometric | |:-------------------------------------------:|:-------:|:-----------:|:--------------:| | Mean | X | X | X | | kth moment | | | | | Variance | X | X | X | | Truncated mean | | | | | Limited mean | | | | | Stop-loss | | | | | Excess of mean | | | | | Moment Generating Function | | X | | | Probability Generating Function | | X | | | Probability Density Function | X | | | | Cumulative Probability Density Function | X | | | | Value-at-Risk | | X | | | Tail Value-at-Risk | | | |

## Copulas

1. Independence Copula
2. Fréchet Lower Bound Copula
3. Fréchet Upper Bound Copula
4. Fréchet Copula
5. Bivariate Gumbel Copula
6. Bivariate Clayton Copula
7. Bivariate Ali-Mikhail-Haq Copula
8. Bivariate Cuadras-Augé Copula
9. Bivariate Marshall-Olkin Copula
10. Bivariate Frank Copula
11. Bivariate Eyraud-Farlie-Gumbel-Morgenstern (EFGM) Copula