poisson_quantile

std.stats.poisson_quantile · Level L5

The Poisson distribution's quantile with mean λ: the smallest whole k with P(K ≤ k) ≥ p. A bisection on whole numbers (a While over poisson_quantile_step), from [0, ⌊λ + t⌋ + 2] with t from Bernstein's inequality for the upper tail.

k = min{ k : Q(k + 1, λ) ≥ p }

Signature

poisson_quantile(p: f64[n], lam: f64[]) → i64[n]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Agrees with the reference k = 0; while exp(-λ)·Σ_{j≤k} λʲ/j! < p: k += 1 to 80 digits (100-digit arithmetic), on all 40 test cases.
All 137 int64 results are exact: the error is zero.
Interpreter and NumPy backend return bit-identical results.

Accuracy in detail

correctly rounded (the int64 nearest the exact value): 100%
bit-equal to the NumPy formula in int64: 100%
largest error, in units in the last place: 0

Note

The reference adds the probabilities one by one until they reach p, so neither the incomplete gamma function nor the bisection checks itself. The answer is exact unless p lies within the CDF's own error of F(k); the verification bounds that case.

Identity

Calls

poisson_bracket_open poisson_quantile_step

Called by

—

sha256:1ab816c8ee62e10d1025af3d2c6213e3ba8aac3387365ae37f000cb9239297ea

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.stats.poisson_quantile · Ω:poisson_quantile
Pins: sha256:5795a95269c6a6ebfeebf5607728a73bf06ccd0dc491683da23178bcbc468867this graph and the 2 it reaches through calls
Evidence: sha256:02b397ded5ab9db52df6aaac3dd3851c7f7745fe806f458d7626bb94762249d2the hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.