CDF of Marginal K Distribution

Computes the cumulative distribution function \(P(K_J \leq k \mid a, b)\) for \(k = 0, 1, \ldots, J\).

Usage

cdf_K_marginal(J, a, b, logS, M = .QUAD_NODES_DEFAULT)

Arguments

J

Integer; sample size.

a

Numeric; shape parameter of Gamma prior (> 0).

b

Numeric; rate parameter of Gamma prior (> 0).

logS

Matrix; pre-computed log-Stirling matrix.

M

Integer; number of quadrature nodes (default: 80).

Value

Numeric vector of length \(J+1\) containing \(P(K_J \leq k \mid a, b)\) for \(k = 0, 1, \ldots, J\).

Details

The CDF satisfies:

• \(F(0) = 0\) (since \(P(K_J = 0) = 0\))

• \(F(J) = 1\)

• \(F(k)\) is non-decreasing in \(k\)

See also

pmf_K_marginal, quantile_K_marginal

Examples

logS <- compute_log_stirling(50)
cdf <- cdf_K_marginal(50, 1.5, 0.5, logS)

# Verify CDF ends at 1
cdf[51]
#> [1] 1

# P(K <= 10)
cdf[11]
#> [1] 0.7009381