NormalMix {EnvStats} | R Documentation |

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)
```

`x` |
vector of quantiles. |

`q` |
vector of quantiles. |

`p` |
vector of probabilities between 0 and 1. |

`n` |
sample size. If |

`mean1` |
vector of means of the first normal random variable.
The default is |

`sd1` |
vector of standard deviations of the first normal random variable.
The default is |

`mean2` |
vector of means of the second normal random variable.
The default is |

`sd2` |
vector of standard deviations of the second normal random variable.
The default is |

`p.mix` |
vector of probabilities between 0 and 1 indicating the mixing proportion.
For |

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)
```

`dnormMix`

gives the density, `pnormMix`

gives the distribution function,
`qnormMix`

gives the quantile function, and `rnormMix`

generates random
deviates.

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.

Steven P. Millard (EnvStats@ProbStatInfo.com)

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.

Normal, LognormalMix, Probability Distributions and Random Numbers.

```
# 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
```

[Package *EnvStats* version 2.8.1 Index]