In the following, let \(X\) be a mixture random variable composed
of \(K\) component distributions \(F_1, F_2, \ldots, F_K\) with
corresponding weights \(w_1, w_2, \ldots, w_K\) where
\(\sum_{i=1}^K w_i = 1\) and \(w_i \geq 0\) for all \(i\).
Probability mixture (type = "probability")
Support: The union of the supports of all component distributions
Mean:
For univariate mixtures:
$$
E(X) = \sum_{i=1}^K w_i \mu_i
$$
where \(\mu_i\) is the mean of the \(i\)-th component distribution.
For multivariate mixtures:
$$
E(\mathbf{X}) = \sum_{i=1}^K w_i \boldsymbol{\mu}_i
$$
where \(\boldsymbol{\mu}_i\) is the mean vector of the \(i\)-th
component distribution.
Variance:
For univariate mixtures:
$$
\text{Var}(X) = \sum_{i=1}^K w_i (\mu_i^2 + \sigma_i^2) - \left(\sum_{i=1}^K w_i \mu_i\right)^2
$$
where \(\sigma_i^2\) is the variance of the \(i\)-th component
distribution.
Covariance:
For multivariate mixtures:
$$
\text{Cov}(\mathbf{X}) = \sum_{i=1}^K w_i \left[ (\boldsymbol{\mu}_i - \bar{\boldsymbol{\mu}})(\boldsymbol{\mu}_i - \bar{\boldsymbol{\mu}})^T + \boldsymbol{\Sigma}_i \right]
$$
where \(\bar{\boldsymbol{\mu}} = \sum_{i=1}^K w_i \boldsymbol{\mu}_i\)
is the overall mean vector and \(\boldsymbol{\Sigma}_i\) is the
covariance matrix of the \(i\)-th component distribution.
Probability density/mass function (p.d.f/p.m.f):
$$
f(x) = \sum_{i=1}^K w_i f_i(x)
$$
where \(f_i(x)\) is the density or mass function of the \(i\)-th
component distribution.
Cumulative distribution function (c.d.f):
For univariate mixtures:
$$
F(x) = \sum_{i=1}^K w_i F_i(x)
$$
where \(F_i(x)\) is the c.d.f. of the \(i\)-th component
distribution.
For multivariate mixtures, the c.d.f. is approximated numerically.
Quantile function:
For univariate probability mixtures, the quantile function has no closed
form and is computed numerically by inverting the c.d.f. using root-finding
(stats::uniroot()).
For multivariate mixtures, quantiles are not yet implemented.
Quantile mixture (type = "quantile")
Also known as a Vincent average or Vincentization, only univariate
component distributions are supported.
Quantile function (closed form):
$$
Q(p) = \sum_{i=1}^K w_i Q_i(p)
$$
where \(Q_i(p)\) is the quantile function of the \(i\)-th component
distribution.
Cumulative distribution function: computed numerically by inverting
\(Q(p)\) via stats::uniroot().
Probability density function: derived analytically from the quantile
function. For \(p = F(x)\):
$$
f(x) = \frac{1}{Q'(p)} = \frac{1}{\sum_{i=1}^K w_i / f_i(Q_i(p))}
$$
Mean: \(E(X) = \sum_{i=1}^K w_i \mu_i\) (identical to the
probability mixture mean).
Variance: computed numerically as
\(\int_0^1 Q(p)^2 \, dp - \left(E(X)\right)^2\).
![[Maturing]](figures/lifecycle-maturing.svg?package=distributional&version=0.9.0)