[Repo Assist] Fix ChiSquared.InvCDF scale parameter bug; add InvCDF tests

src/FSharp.Stats/Distributions/Continuous/ChiSquared.fs

@@ -140,8 +140,11 @@
     /// <param name="p">The probability value in [0.0, 1.0].</param>
     /// <returns>The quantile corresponding to the cumulative probability p.</returns>
     static member InvCDF (dof: float) (p: float) : float =
+        // Chi-squared(dof) = Gamma(alpha=dof/2, scale=2).
+        // FSharp.Stats Gamma uses the scale parameterisation (Mean = alpha * scale),
+        // so the correct scale is 2.0, not 1/2.
         let alpha = dof / 2.0
-        let beta  = 1. / 2.0
+        let beta  = 2.0
         Gamma.InvCDF alpha beta p
 
     /// <summary>Returns the support of the exponential distribution: [0, Positive Infinity).</summary>
@@ -199,7 +202,7 @@
     /// </example>
     static member Init dof =
         { new ContinuousDistribution<float,float> with
             member d.Mean              = ChiSquared.Mean dof
             member d.StandardDeviation = ChiSquared.StandardDeviation dof 
             member d.Variance          = ChiSquared.Variance dof
             member d.CDF x             = ChiSquared.CDF dof  x 

tests/FSharp.Stats.Tests/DistributionsContinuous.fs

@@ -618,6 +618,65 @@ let chiSquaredTests =
                 Expect.floatClose Accuracy.veryHigh (testCase.PDF -1.) 0. "Should be equal"
                 Expect.isTrue (Ops.isNan <| testCase.PDF nan) "Should be equal"
         ]
+
+        // Tests for ChiSquared.InvCDF (the percentile-point function).
+        // Reference values from standard chi-square tables (and CDF values already tested above).
+        testList "ChiSquared.InvCDF tests" [
+
+            test "ChiSquared.InvCDF returns 0.0 for p=0.0" {
+                let x = Continuous.ChiSquared.InvCDF 5. 0.
+                Expect.floatClose Accuracy.high x 0. "InvCDF(p=0) should be 0"
+            }
+
+            test "ChiSquared.InvCDF returns +∞ for p=1.0" {
+                let x = Continuous.ChiSquared.InvCDF 5. 1.
+                Expect.isTrue (x = infinity) "InvCDF(p=1) should be +infinity"
+            }
+
+            // Round-trip tests: CDF(InvCDF(p)) ≈ p
+            test "ChiSquared.InvCDF round-trip dof=5 p=0.5" {
+                let dof = 5.
+                let p   = 0.5
+                let x   = Continuous.ChiSquared.InvCDF dof p
+                let p2  = Continuous.ChiSquared.CDF dof x
+                Expect.floatClose Accuracy.high p p2 "CDF(InvCDF(0.5)) ≈ 0.5"
+            }
+
+            test "ChiSquared.InvCDF round-trip dof=1 p=0.95" {
+                let dof = 1.
+                let p   = 0.95
+                let x   = Continuous.ChiSquared.InvCDF dof p
+                let p2  = Continuous.ChiSquared.CDF dof x
+                Expect.floatClose Accuracy.high p p2 "CDF(InvCDF(0.95)) ≈ 0.95"
+            }
+
+            test "ChiSquared.InvCDF round-trip dof=10 p=0.01" {
+                let dof = 10.
+                let p   = 0.01
+                let x   = Continuous.ChiSquared.InvCDF dof p
+                let p2  = Continuous.ChiSquared.CDF dof x
+                Expect.floatClose Accuracy.high p p2 "CDF(InvCDF(0.01)) ≈ 0.01"
+            }
+
+            // Known-value tests derived from the CDF cases tested above
+            // (Williams 1984 table: CDF(20, 12.443) ≈ 0.1  →  InvCDF(20, 0.1) ≈ 12.443)
+            test "ChiSquared.InvCDF known value dof=20 p=0.1" {
+                let x = Continuous.ChiSquared.InvCDF 20. 0.1
+                Expect.floatClose Accuracy.low x 12.443 "InvCDF(20, 0.1) should be ≈ 12.443"
+            }
+
+            // CDF(3, 1.424) ≈ 0.3  →  InvCDF(3, 0.3) ≈ 1.424
+            test "ChiSquared.InvCDF known value dof=3 p=0.3" {
+                let x = Continuous.ChiSquared.InvCDF 3. 0.3
+                Expect.floatClose Accuracy.low x 1.424 "InvCDF(3, 0.3) should be ≈ 1.424"
+            }
+
+            // CDF(100, 129.561) ≈ 0.975  →  InvCDF(100, 0.975) ≈ 129.561
+            test "ChiSquared.InvCDF known value dof=100 p=0.975" {
+                let x = Continuous.ChiSquared.InvCDF 100. 0.975
+                Expect.floatClose Accuracy.low x 129.561 "InvCDF(100, 0.975) should be ≈ 129.561"
+            }
+        ]
     ]
 
 //Test ommitted due to long runtime of CodeCov