NormalMix: Mixture of Two Normal Distributions

Description

Density, distribution function, quantile function, and random generation for a mixture of two normal distribution with parameters mean1, sd1, mean2, sd2, and p.mix.

Usage

dnormMix(x, mean1 = 0, sd1 = 1, mean2 = 0, sd2 = 1, p.mix = 0.5)
  pnormMix(q, mean1 = 0, sd1 = 1, mean2 = 0, sd2 = 1, p.mix = 0.5)
  qnormMix(p, mean1 = 0, sd1 = 1, mean2 = 0, sd2 = 1, p.mix = 0.5)
  rnormMix(n, mean1 = 0, sd1 = 1, mean2 = 0, sd2 = 1, p.mix = 0.5)

Arguments

vector of quantiles.

vector of probabilities between 0 and 1.

sample size. If length(n) is larger than 1, then length(n) random values are returned.

mean1

vector of means of the first normal random variable. The default is mean1=0.

sd1

vector of standard deviations of the first normal random variable. The default is sd1=1.

mean2

vector of means of the second normal random variable. The default is mean2=0.

sd2

vector of standard deviations of the second normal random variable. The default is sd2=1.

p.mix

vector of probabilities between 0 and 1 indicating the mixing proportion. For rnormMix this must be a single, non-missing number.

Value

dnormMix gives the density, pnormMix gives the distribution function, qnormMix gives the quantile function, and rnormMix generates random deviates.

Details

Let $f(x; \mu, \sigma)$ denote the density of a normal random variable with parameters mean=$\mu$ and sd=$\sigma$. The density, $g$, of a normal mixture random variable with parameters mean1=$\mu_1$, sd1=$\sigma_1$, mean2=$\mu_2$, sd2=$\sigma_2$, and p.mix=$p$ is given by:

g (x; μ_{1}, σ_{1}, μ_{2}, σ_{2}, p) = (1 - p) f (x; μ_{1}, σ_{1}) + p f (x; μ_{2}, σ_{2})

References

Johnson, N. L., S. Kotz, and A.W. Kemp. (1992). Univariate Discrete Distributions. Second Edition. John Wiley and Sons, New York, pp.53-54, and Chapter 8. Johnson, N. L., S. Kotz, and N. Balakrishnan. (1994). Continuous Univariate Distributions, Volume 1. Second Edition. John Wiley and Sons, New York.

Examples

Run this code

# Density of a normal mixture with parameters mean1=0, sd1=1, 
  #  mean2=4, sd2=2, p.mix=0.5, evaluated at 1.5: 

  dnormMix(1.5, mean2=4, sd2=2) 
  #[1] 0.1104211

  #----------

  # The cdf of a normal mixture with parameters mean1=10, sd1=2, 
  # mean2=20, sd2=2, p.mix=0.1, evaluated at 15: 

  pnormMix(15, 10, 2, 20, 2, 0.1) 
  #[1] 0.8950323

  #----------

  # The median of a normal mixture with parameters mean1=10, sd1=2, 
  # mean2=20, sd2=2, p.mix=0.1: 

  qnormMix(0.5, 10, 2, 20, 2, 0.1) 
  #[1] 10.27942

  #----------

  # Random sample of 3 observations from a normal mixture with 
  # parameters mean1=0, sd1=1, mean2=4, sd2=2, p.mix=0.5. 
  # (Note: the call to set.seed simply allows you to reproduce this example.)

  set.seed(20) 
  rnormMix(3, mean2=4, sd2=2)
  #[1] 0.07316778 2.06112801 1.05953620

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