Description Usage Arguments Details Value Note Author(s) References See Also Examples

Density, distribution function, quantile function, and random generation
for a mixture of two normal distribution with parameters
`mean1`

, `sd1`

, `mean2`

, `sd2`

, and `p.mix`

.

1 2 3 4 |

`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; μ, σ)* denote the density of a
normal random variable with parameters
`mean=`

*μ* and `sd=`

*σ*. The density, *g*, of a
normal mixture random variable with parameters `mean1=`

*μ_1*,
`sd1=`

*σ_1*, `mean2=`

*μ_2*,
`sd2=`

*σ_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)*

`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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 | ```
# 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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