MultivariatePoissonDistribution

Wolfram's common-shock model — $X_i = Y_0 + Y_i$ , a shared $Y_0 \sim Poisson(\mu_0)$ plus an independent $Y_i \sim Poisson(\mu_i)$ per coordinate.

MultivariatePoissonDistribution(mu0, {mu1, ..., muk})shared shock mu0 plus per-coordinate independent Poisson(mu_i).

Typed signatures

Type	Package	Applies
`(real<0..>, list<real>) -> distribution`	`statistics`	always

Domain: Statistics

Details

$PDF(x) = \sum_{j=0}^{\min_i x_i} PDF(Poisson(\mu_0), j) \prod_i PDF(Poisson(\mu_i), x_i-j)$ — a finite EXACT sum built from compute-engine's own native Poisson PDF, needing every $x_i$ to resolve to a concrete nonnegative integer (symbolic $x_i$ stays unevaluated).
$Mean_i = \mu_0 + \mu_i$ ; $Variance_i = \mu_0 + \mu_i$ (Poisson's own mean-equals-variance identity carries over, since each $X_i$ is a sum of independent Poissons).
Covariance (compute-engine native over two collections) is widened to also accept a single MultivariatePoissonDistribution: $Cov(X_i,X_j) = \mu_0$ ( $i \ne j$ , from the shared shock), $Var(X_i) = \mu_0+\mu_i$ on the diagonal.
RandomVariate draws $Y_0 \sim Poisson(\mu_0)$ once, then $X_i = Y_0 + Y_i$ with each $Y_i \sim Poisson(\mu_i)$ independent — seeded through the same SeedRandom stream every other distribution in this package uses.

Examples

Basic #

\mathrm{mean}(\mathrm{MultivariatePoissonDistribution}(1, \bigl\lbrack2, 3\bigr\rbrack))

Out

\bigl\lbrack3, 4\bigr\rbrack

matches Mean, exactly Poisson's own identity.

\mathrm{Variance}(\mathrm{MultivariatePoissonDistribution}(1, \bigl\lbrack2, 3\bigr\rbrack))

Out

\bigl\lbrack3, 4\bigr\rbrack

off-diagonal is mu0 = 1, the shared shock.

\operatorname{Covariance}(\mathrm{MultivariatePoissonDistribution}(1, \bigl\lbrack2, 3\bigr\rbrack))

Out

\bigl\lbrack\bigl\lbrack3, 1\bigr\rbrack, \bigl\lbrack1, 4\bigr\rbrack\bigr\rbrack

\operatorname{PDF}(\mathrm{MultivariatePoissonDistribution}(1, \bigl\lbrack2, 3\bigr\rbrack), \bigl\lbrack1, 1\bigr\rbrack)

Out

\frac{7}{e^6}

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Examples	4