MixtureDistribution

The weighted mixture of $d_1, \ldots, d_n$ with (renormalized) weights $w_1, \ldots, w_n$ .

MixtureDistribution({w1, ..., wn}, {d1, ..., dn})the weighted mixture of d1, ..., dn; weights are normalized by their sum.

Typed signatures

Type	Package	Applies
`(list<real>, list<distribution>) -> distribution`	`statistics`	always

Domain: Statistics

Details

Weights are normalized by their sum before use, so $\{1,1\}$ and $\{1,2\}$ -scaled-by-3 agree.
$PDF(x) = \sum_i (w_i/\sum w) \cdot PDF(d_i,x)$ ; $CDF$ the same weighted sum of $CDF(d_i,x)$ .
$Mean = \sum_i (w_i/\sum w) \cdot Mean(d_i)$ .
$Variance$ by the law of total variance: $\sum_i \hat w_i \, Var(d_i) + \sum_i \hat w_i (Mean(d_i) - Mean)^2$ , where $\hat w_i$ is the normalized weight.
RandomVariate picks a component by its normalized weight against one uniform draw, then delegates to RandomVariate on that component — so it reuses whatever per-kind sampler that component already has, old or new, including another compound distribution.

Examples

Basic #

\mathrm{mean}(\mathrm{MixtureDistribution}(\bigl\lbrack1, 1\bigr\rbrack, \bigl\lbrack\operatorname{NormalDistribution}(0, 1), \operatorname{NormalDistribution}(10, 1)\bigr\rbrack))

Out

5

\mathrm{Variance}(\mathrm{MixtureDistribution}(\bigl\lbrack1, 1\bigr\rbrack, \bigl\lbrack\operatorname{NormalDistribution}(0, 1), \operatorname{NormalDistribution}(10, 1)\bigr\rbrack))

Out

26

N(\operatorname{CDF}(\mathrm{MixtureDistribution}(\bigl\lbrack1, 1\bigr\rbrack, \bigl\lbrack\operatorname{NormalDistribution}(0, 1), \operatorname{NormalDistribution}(10, 1)\bigr\rbrack), 100))

Out

1.0

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Examples	3