dist.Normal.Mixture {LaplacesDemon} | R Documentation |

## Mixture of Normal Distributions

### Description

These functions provide the density, cumulative, and random generation
for the mixture of univariate normal distributions with probability
`p`

, mean `\mu`

and standard deviation `\sigma`

.

### Usage

```
dnormm(x, p, mu, sigma, log=FALSE)
pnormm(q, p, mu, sigma, lower.tail=TRUE, log.p=FALSE)
rnormm(n, p, mu, sigma)
```

### Arguments

`x` , `q` |
This is vector of values at which the density will be evaluated. |

`p` |
This is a vector of length |

`n` |
This is the number of observations, which must be a positive integer that has length 1. |

`mu` |
This is a vector of length |

`sigma` |
This is a vector of length |

`lower.tail` |
Logical. This defaults to |

`log` , `log.p` |
Logical. If |

### Details

Application: Continuous Univariate

Density:

`p(\theta) = \sum p_i \mathcal{N}(\mu_i, \sigma^2_i)`

Inventor: Unknown

Notation 1:

`\theta \sim \mathcal{N}(\mu, \sigma^2)`

Notation 2:

`p(\theta) = \mathcal{N}(\theta | \mu, \sigma^2)`

Parameter 1: mean parameters

`\mu`

Parameter 2: standard deviation parameters

`\sigma > 0`

Mean:

`E(\theta) = \sum p_i \mu_i`

Variance:

`var(\theta) = \sum p_i \sigma^{0.5}_i`

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 normal mixture, or Gaussian mixture, distribution is a combination of normal probability distributions.

### Value

`dnormm`

gives the density,
`pnormm`

returns the CDF, and
`rnormm`

generates random deviates.

### Author(s)

Statisticat, LLC. software@bayesian-inference.com

### See Also

`ddirichlet`

and
`dnorm`

.

### Examples

```
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, dnormm(x, p, mu, sigma, log=FALSE), type="l") #Density
plot(x, pnormm(x, p, mu, sigma), type="l") #CDF
plot(density(rnormm(10000, p, mu, sigma))) #Random Deviates
```

*LaplacesDemon*version 16.1.6 Index]