Journal of Statistical Distributions and Applications

Table 4 PMF of trivariate binary variables

From: Multivariate distributions of correlated binary variables generated by pair-copulas

(Y₁,Y₂,Y₃)	Probability
(0, 0, 0)	q₂∗C_13\|0(q_1\|0,q_3\|0)
(0, 0, 1)	C₁₂(q₁,q₂)−q₂∗C_13\|0(q_1\|0,q_3\|0)
(0, 1, 0)	p₂∗C_13\|1(q_1\|1,q_3\|1)
(0, 1, 1)	q₁−C₁₂(q₁,q₂)−p₂∗C_13\|1(q_1\|1,q_3\|1)
(1, 0, 0)	C₂₃(q₂,q₃)−q₂∗C_13\|0(q_1\|0,q_3\|0)
(1, 0, 1)	q₂−C₂₃(q₂,q₃)−C₁₂(q₁,q₂)+q₂∗C_13\|0(q_1\|0,q_3\|0)
(1, 1, 0)	q₃−C₂₃(q₂,q₃)−p₂∗C_13\|1(q_1\|1,q_3\|1)
(1, 1, 1)	1−q₁−q₂−q₃+C₁₂(q₁,q₂)+C₂₃(q₂,q₃)+p₂∗C_13\|1(q_1\|1,q_3\|1)

Note: There are 7 parameters here: marginal means p₁,p₂,p₃ and copula parameters \(\phantom {\dot {i}\!}\theta _{12},\theta _{23}, \theta _{13|Y_{2}=0}\), and \(\phantom {\dot {i}\!}\theta _{13|Y_{2}=1}\)

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