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

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

 ```1 2 3``` ```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 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. `mu` This is a vector of length M that is the mean parameter mu. `sigma` This is a vector of length M that is the standard deviation parameter sigma, which must be positive. `lower.tail` Logical. This defaults to `TRUE`. `log,log.p` Logical. If `TRUE`, then probabilities p are given as log(p).

Details

• Application: Continuous Univariate

• Density: p(theta) = sum p[i] N(mu[i], sigma[i]^2)

• Inventor: Unknown

• Notation 1: theta ~ N(mu, sigma^2)

• Notation 2: p(theta) = 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[i]^(0.5)

• 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. [email protected]

`ddirichlet` and `dnorm`.
 ```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, 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 ```