Binomial count and iid values in fixed-budget rejection sampling
= Binomial count and iid values in fixed-budget rejection sampling
{title2=$N\sim\operatorname{Bin}(n,1/M)$}
For n independent proposals and an envelope constant $M$, each rejection decision succeeds with probability $1/M$. The accepted count is Bin(n,1/M). Conditional on any acceptance pattern, the accepted values have independent target density f; summing over patterns preserves that product law conditional on the count. The empty-output probability is $(1-1/M)^n$.