Multivariate Probability Distributions.
This package contains Java classes providing methods to compute mass, density, distribution and complementary distribution functions for some multi-dimensional discrete and continuous probability distributions. It does not generate random numbers for multivariate distributions; for that, see the package umontreal.ssj.randvarmulti.
DefinitionsWe recall that the distribution function of a continuous random vector \(X= \{x_1, x_2, …, x_d\}\) with density \(f(x_1, x_2, …, x_d)\) over the \(d\)-dimensional space \(R^d\) is
\begin{align} F(x_1, x_2, …, x_d) & = P[X_1\le x_1, X_2\le x_2, …, X_d\le x_d] \\ & = \int_{-\infty}^{x_1}\int_{-\infty}^{x_2} \cdots\int_{-\infty}^{x_d} f(s_1, s_2, …, s_d)\; ds_1 ds_2 …ds_d \tag{FDist} \end{align}
while that of a discrete random vector \(X\) with mass function \(\{p_1, p_2, …, p_d\}\) over a fixed set of real numbers is
\begin{align} F(x_1, x_2, …, x_d) & = P[X_1\le x_1, X_2\le x_2, …, X_d\le x_d] \\ & = \sum_{i_1\le x_1}\sum_{i_2\le x_2} \cdots\sum_{i_d\le x_d} p(x_1, x_2, …, x_d), \tag{FDistDisc} \end{align}
where \(p(x_1, x_2, …, x_d) = P[X_1 = x_1, X_2 = x_2, …, X_d = x_d]\). For a discrete distribution over the set of integers, one has
\begin{align} F (x_1, x_2, …, x_d) & = P[X_1\le x_1, X_2\le x_2, …, X_d\le x_d] \\ & = \sum_{s_1=-\infty}^{x_1} \sum_{s_2=-\infty}^{x_2} \cdots\sum_{s_d=-\infty}^{x_d} p(s_1, s_2, …, s_d), \tag{FDistDiscInt} \end{align}
where \(p(s_1, s_2, …, s_d) = P[X_1=s_1, X_2=s_2, …, X_d=s_d]\).
We define \(\bar{F}\), the complementary distribution function of \(X\), as
\[ \bar{F} (x_1, x_2, …, x_d) = P[X_1\ge x_1, X_2\ge x_2, …, X_d\ge x_d]. \]
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