Generates random samples from the w₁ distribution by first sampling
α ~ Gamma(a, b), then w₁ | α ~ Beta(1, α).
Arguments
- n
Integer; number of samples to generate.
- a
Numeric; shape parameter of the Gamma prior on α (a > 0).
- b
Numeric; rate parameter of the Gamma prior on α (b > 0).
Value
Numeric vector of length n; random samples from the w₁ distribution.
Details
This function uses the hierarchical representation:
α ~ Gamma(a, b)
w₁ | α ~ Beta(1, α)
Useful for Monte Carlo validation of the closed-form functions.
Examples
# Generate samples
set.seed(42)
samples <- rw1(10000, a = 2, b = 1)
# Compare empirical vs theoretical mean
mean(samples) # ~0.404
#> [1] 0.4023996
mean_w1(a = 2, b = 1) # 0.4037
#> [1] 0.4036526
