student_t_cdf

std.stats.student_t_cdf · Level L5

Student's t distribution's cumulative probability with ν degrees of freedom, through the incomplete beta function: ½·I_{ν/(ν+x²)}(ν/2, ½) on the left, one minus that on the right.

x ≤ 0: ½·I_{ν/(ν+x²)}(ν/2, ½); x > 0: 1 − that

Signature

student_t_cdf(x: f64[n], nu: f64[]) → 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]nuf64[]0.50.0Multiplyx2MultiplyaGreaterrightAddnux2DividetBetaInciMultiplytail1.0SubtractupperWhereppf64[n]
  • input
  • operation
  • constant
  • call
  • output

Verification

  • Signature proven by NOVA’s shape solver, for every size.
  • Agrees with the reference ½ + x·Γ((ν+1)/2)·₂F₁(½, (ν+1)/2; 3/2; −x²/ν)/(√(πν)·Γ(ν/2)) to 80 digits (100-digit arithmetic), on all 40 test cases.
  • All 160 float64 results inside the running error bound; the closest uses 64% of it.
  • Interpreter and NumPy backend return bit-identical results.
Accuracy in detail
correctly rounded (the float64 nearest the exact value)
29%
bit-equal to the NumPy formula in float64
100%
largest error, in units in the last place
3.5e+6

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

The reference is the hypergeometric form of the same CDF, a different special function from the incomplete beta the graph uses.

Identity

Calls
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Called by
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sha256:45ba351eac230cf1d8bd17696b37fd2e845481c46fbd8451c047d7cac0f7ddc7

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