# dist.Laplace.Mixture: Mixture of Laplace Distributions In LaplacesDemon: Complete Environment for Bayesian Inference

## Description

These functions provide the density, cumulative, and random generation for the mixture of univariate Laplace distributions with probability p, location mu and scale sigma.

## Usage

 ```1 2 3``` ```dlaplacem(x, p, location, scale, log=FALSE) plaplacem(q, p, location, scale) rlaplacem(n, p, location, scale) ```

## Arguments

 `x,q` This is vector of values at which the density will be evaluated. `p` This is a vector of length M of probabilities for M components. The sum of the vector must be one. `n` This is the number of observations, which must be a positive integer that has length 1. `location` This is a vector of length M that is the location parameter mu. `scale` This is a vector of length M that is the scale parameter sigma, which must be positive. `log` Logical. If `TRUE`, then the logarithm of the density is returned.

## Details

• Application: Continuous Univariate

• Density: p(theta) = sum p[i] L(mu[i], sigma[i])

• Inventor: Unknown

• Notation 1: theta ~ L(mu, sigma)

• Notation 2: p(theta) = L(theta | mu, sigma)

• Parameter 1: location parameters mu

• Parameter 2: scale parameters sigma > 0

• Mean: E(theta) = sum p[i] mu[i]

• Variance:

• Mode:

A mixture distribution is a probability distribution that is a combination of other probability distributions, and each distribution is called a mixture component, or component. A probability (or weight) exists for each component, and these probabilities sum to one. A mixture distribution (though not these functions here in particular) may contain mixture components in which each component is a different probability distribution. Mixture distributions are very flexible, and are often used to represent a complex distribution with an unknown form. When the number of mixture components is unknown, Bayesian inference is the only sensible approach to estimation.

A Laplace mixture distribution is a combination of Laplace probability distributions.

One of many applications of Laplace mixture distributions is the Laplace Mixture Model (LMM).

## Value

`dlaplacem` gives the density, `plaplacem` returns the CDF, and `rlaplacem` generates random deviates.

## Author(s)

Statisticat, LLC. software@bayesian-inference.com

`ddirichlet` and `dlaplace`.

## Examples

 ```1 2 3 4 5 6 7 8``` ```library(LaplacesDemon) p <- c(0.3,0.3,0.4) mu <- c(-5, 1, 5) sigma <- c(1,2,1) x <- seq(from=-10, to=10, by=0.1) plot(x, dlaplacem(x, p, mu, sigma, log=FALSE), type="l") #Density plot(x, plaplacem(x, p, mu, sigma), type="l") #CDF plot(density(rlaplacem(10000, p, mu, sigma))) #Random Deviates ```

### Example output

```
```

LaplacesDemon documentation built on July 9, 2021, 5:07 p.m.