EnvStats (version 2.7.0)

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)

Value

dnormMix gives the density, pnormMix gives the distribution function,

qnormMix gives the quantile function, and rnormMix generates random deviates.

Arguments

x

vector of quantiles.

q

vector of quantiles.

p

vector of probabilities between 0 and 1.

n

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.

Author

Steven P. Millard ()

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: $$g(x; \mu_1, \sigma_1, \mu_2, \sigma_2, p) = (1 - p) f(x; \mu_1, \sigma_1) + p f(x; \mu_2, \sigma_2)$$

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.

See Also

Normal, LognormalMix, Probability Distributions and Random Numbers.

Examples

  # 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