Mixture of Two Normal Distributions

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

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

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.

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

Value

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

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.

Author

Steven P. Millard (EnvStats@ProbStatInfo.com)

Note

A normal mixture distribution is sometimes used to model data that appear to be “contaminated”; that is, most of the values appear to come from a single normal distribution, but a few “outliers” are apparent. In this case, the value of mean2 would be larger than the value of mean1, and the mixing proportion p.mix would be fairly close to 0 (e.g., p.mix=0.1). The value of the second standard deviation (sd2) may or may not be the same as the value for the first (sd1).

Another application of the normal mixture distribution is to bi-modal data; that is, data exhibiting two modes.

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
  #[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
  #[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
  #[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
  #[1] 0.07316778 2.06112801 1.05953620