BCD (version 0.1.1)

dbinomBCD: Joint Probability Mass Function for a Bivariate Binomial Distribution via Conditional Specification

Description

Computes the probability mass function (p.m.f.) of the bivariate binomial conditionals distribution (BBCD) as defined by Ghosh, Marques, and Chakraborty (2025). The distribution is characterized by conditional binomial distributions for \( X \) and \( Y \).

Usage

dbinomBCD(x, y, n1, n2, p1, p2, lambda)

Value

The probability \( P(X = x, Y = y)\).

Arguments

x

value of \( X \), must be in \(\{0, 1, ..., n_1\}\)

y

value of \( Y \), must be in \(\{0, 1, ..., n_2\}\)

n1

number of trials for \( X \), must be non-negative

n2

number of trials for \( Y \), must be non-negative

p1

base success probability for \( X \), in \((0, 1)\)

p2

base success probability for \( Y \), in \((0, 1)\)

lambda

dependence parameter, must be positive.

Details

The joint p.m.f. of the BBCD is $$ P(X = x, Y = y) = K_B(n_1, n_2, p_1, p_2, \lambda) \binom{n_1}{x} \binom{n_2}{y} p_1^x p_2^y (1 - p_1)^{n_1 - x} (1 - p_2)^{n_2 - y} \lambda^{xy}, $$ where \( x = 0, 1, \ldots, n_1 \), \( y = 0, 1, \ldots, n_2 \), and \( K_B(n_1, n_2, p_1, p_2, \lambda) \) is the normalizing constant.

References

Ghosh, I., Marques, F., & Chakraborty, S. (2025). A form of bivariate binomial conditionals distributions. Communications in Statistics - Theory and Methods, 54(2), 534--553. tools:::Rd_expr_doi("10.1080/03610926.2024.2315294")

See Also

pbinomBCD rbinomBCD MLEbinomBCD

Examples

# Compute P(X = 2, Y = 1) with n1 = 5, n2 = 5, p1 = 0.5, p2 = 0.4, lambda = 0.5
dbinomBCD(x = 2, y = 1, n1 = 5, n2 = 5, p1 = 0.5, p2 = 0.4, lambda = 0.5)

# Example with independence (lambda = 1)
dbinomBCD(x = 2, y = 1, n1 = 5, n2 = 5, p1 = 0.5, p2 = 0.4, lambda = 1.0)