chi2_cdf

std.stats.chi2_cdf · Level L5

The chi-square distribution's cumulative probability with k degrees of freedom: P(k/2, x/2), the regularized lower incomplete gamma function.

F(x; k) = P(k/2, x/2)

Signature

chi2_cdf(x: f64[n], k: i64[]) → f64[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.

xf64[n]0.5ki64[]MultiplytMultiplyaGammaIncppf64[n]
  • input
  • operation
  • constant
  • call
  • output

Verification

  • Signature proven by NOVA’s shape solver, for every size.
  • Agrees with the reference closed forms for whole k: 1 − e^(−x/2)·Σ…, or erf(√(x/2)) − … to 80 digits (100-digit arithmetic), on all 40 test cases.
  • All 192 float64 results inside the running error bound; the closest uses 6% of it.
  • Interpreter and NumPy backend return bit-identical results.
Accuracy in detail
correctly rounded (the float64 nearest the exact value)
13%
bit-equal to the NumPy formula in float64
100%
largest error, in units in the last place
64

Large ulp counts appear where a result is tiny next to the numbers it is computed from (after cancellation, for example), so one unit in the last place is tiny too; the absolute error is still inside the bound. Results within their own error of zero are not counted.

Note

For whole degrees of freedom the chi-square CDF has closed forms, through finite sums and erf. The reference uses them, so the incomplete gamma function is checked against a different route.

Identity

Calls
—
Called by
—
sha256:385c71a5f2a97ee042d9b3be634b4069895f30e24fd4b4cc9a7fd8bb97e2c6eb

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.chi2_cdf · Ω:chi2_cdf
Pins
sha256:1b67d5f176659ceda21866c5ef78fe28848ab9fb3a317e6ea493d3b27045523bthis graph alone
Evidence
sha256:44b8ba6faabbcecf448e293c658f18b0355a5ff122262f7b5fa8cf7671af587ethe 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.