{"schema_version":"dataset-v1","id":"chernoff-density-curve","version":1,"title":"The Chernoff density","description":"The standard Chernoff density f of Z = argmax(W(t) − t²), computed from the cited Airy factorisation f(x) = C_f g(x) g(−x) with C_f = 2^(−1/3)/Ai(0)², and its negative log-curvature κ(x) = −(log f)″(x).","license":"CC0-1.0 (computed values)","problems":["chernoff-density"],"statements":[{"problem":"chernoff-density","statement":"chernoff-airy-factor","role":"The Airy factorisation used to compute f."},{"problem":"chernoff-density","statement":"chernoff-ordinary-log-concavity","role":"f is log-concave (2014)."}],"figure":{"family":"line-plot","title":"The Chernoff density","caption":"The density of the location Z where W(t) − t² is largest: symmetric, with its mode at 0 and very light tails.","x_label":"x","y_label":"density f(x)","series":["density"],"boundary":"A numerical computation from the cited Airy formula, not a proof; plotted on [−2, 2].","statement":"chernoff-airy-factor"},"provenance":{"method":"g and its first two derivatives by numerical inversion of the bilateral Laplace transform M(s) = Ai(0)/Ai(2^(−1/3)s) along the imaginary axis (mpmath quad, 20 digits) at x = −3, −2.95, …, 3. Checks: Simpson mass 1.000000000, symmetry, and κ(x) > 4|x| at every grid point; variance 0.2635596.","software":{"python":"3.12.15","mpmath":"1.3.0"},"precision":"20-digit quadrature, stored as floating point; a numerical illustration.","generator":"openmath/scripts/build-datasets.py","generated_on":"2026-10-21"},"data":{"series":[{"key":"density","label":"f(x)","x":[-2,-1.95,-1.9,-1.85,-1.8,-1.75,-1.7,-1.65,-1.6,-1.55,-1.5,-1.45,-1.4,-1.35,-1.3,-1.25,-1.2,-1.15,-1.1,-1.05,-1,-0.95,-0.9,-0.85,-0.8,-0.75,-0.7,-0.65,-0.6,-0.55,-0.5,-0.45,-0.4,-0.35,-0.3,-0.25,-0.2,-0.15,-0.1,-0.05,0,0.05,0.1,0.15,0.2,0.25,0.3,0.35,0.4,0.45,0.5,0.55,0.6,0.65,0.7,0.75,0.8,0.85,0.9,0.95,1,1.05,1.1,1.15,1.2,1.25,1.3,1.35,1.4,1.45,1.5,1.55,1.6,1.65,1.7,1.75,1.8,1.85,1.9,1.95,2],"y":[0.00012120178870205772,0.00020221232115409264,0.0003306234539880364,0.0005300138685294316,0.0008334261075335775,0.0012860877476233302,0.0019484691522050437,0.002899546217510499,0.0042400642026900375,0.006095524792684003,0.008618552068725744,0.011990244614230336,0.016420101295877156,0.0221441266339473,0.02942078431516652,0.03852457596010946,0.04973717290440272,0.06333621196553729,0.07958206736317773,0.09870311193298238,0.12088016174036093,0.14623094017763238,0.17479548476682405,0.20652344142622703,0.2412641424904048,0.2787602487036934,0.3186455605185259,0.36044738429797246,0.4035935918833471,0.44742425633591937,0.4912075008593104,0.5341589780715826,0.5754642155322188,0.6143029291364214,0.6498743227120426,0.6814223599385739,0.7082600103238954,0.7297915288603598,0.7455319222412176,0.7551228758409161,0.7583445580537332,0.7551228758409161,0.7455319222412176,0.7297915288603598,0.7082600103238954,0.6814223599385739,0.6498743227120426,0.6143029291364214,0.5754642155322188,0.5341589780715826,0.4912075008593104,0.44742425633591937,0.4035935918833471,0.36044738429797246,0.3186455605185259,0.2787602487036934,0.2412641424904048,0.20652344142622703,0.17479548476682405,0.14623094017763238,0.12088016174036093,0.09870311193298238,0.07958206736317773,0.06333621196553729,0.04973717290440272,0.03852457596010946,0.02942078431516652,0.0221441266339473,0.016420101295877156,0.011990244614230336,0.008618552068725744,0.006095524792684003,0.0042400642026900375,0.002899546217510499,0.0019484691522050437,0.0012860877476233302,0.0008334261075335775,0.0005300138685294316,0.0003306234539880364,0.00020221232115409264,0.00012120178870205772]}],"marks":[]},"sha256":"86d132590c726c3d42e8abe633c71ce1e10d4e08852e0279ba14c24f5b6571f4"}