{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The probability density function for the full Cauchy distribution is" } ] }, { "__type": "Math", "__tag": 4058, "value": "P(x; x_0, \\gamma) = \\frac{1}{\\pi \\gamma \\bigl[ 1+(\\frac{x-x_0}{\\gamma})^2 \\bigr] }" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "and the Standard Cauchy distribution just sets " }, { "__type": "InlineMath", "__tag": 4057, "value": "x_0=0" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\gamma=1" } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The Cauchy distribution arises in the solution to the driven harmonic oscillator problem, and also describes spectral line broadening. It also describes the distribution of values at which a line tilted at a random angle will cut the x axis." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "When studying hypothesis tests that assume normality, seeing how the tests perform on data from a Cauchy distribution is a good indicator of their sensitivity to a heavy-tailed distribution, since the Cauchy looks very much like a Gaussian distribution, but with heavier tails." } ] } ], "title": [], "level": 0, "target": null }, "Warns": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Raises": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Yields": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Methods": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Returns": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "samples", "annotation": "ndarray or scalar", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The drawn samples." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Draw samples from a standard Cauchy distribution with mode = 0." } ] } ], "title": [], "level": 0, "target": null }, "Receives": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Warnings": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Attributes": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Parameters": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "size", "annotation": "int or tuple of ints, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Output shape. If the given shape is, e.g., " }, { "__type": "InlineCode", "__tag": 4051, "value": "(m, n, k)" }, { "__type": "Text", "__tag": 4046, "value": ", then " }, { "__type": "InlineCode", "__tag": 4051, "value": "m * n * k" }, { "__type": "Text", "__tag": 4046, "value": " samples are drawn. Default is None, in which case a single value is returned." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Also known as the Lorentz distribution." } ] }, { "__type": "Admonition", "__tag": 4056, "kind": "note", "base_type": "note", "children": [ { "__type": "AdmonitionTitle", "__tag": 4055, "children": [ { "__type": "Text", "__tag": 4046, "value": "note " } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "New code should use the " }, { "__type": "CrossRef", "__tag": 4002, "value": "standard_cauchy", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "numpy", "version": "*", "kind": "api", "path": "numpy.random._generator:Generator.standard_cauchy" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " method of a " }, { "__type": "CrossRef", "__tag": 4002, "value": "Generator", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "numpy", "version": "*", "kind": "api", "path": "numpy.random._generator:Generator" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " instance instead; please see the " }, { "__type": "InlineRole", "__tag": 4003, "value": "random-quick-start", "domain": null, "role": "ref", "inventory": null }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] } ], "title": [], "level": 0, "target": null }, "Other Parameters": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null } }, "_ordered_sections": [ "Summary", "Extended Summary", "Parameters", "Attributes", "Methods", "Returns", "Yields", "Receives", "Other Parameters", "Raises", "Warns", "Warnings", "Notes" ], "item_file": null, "item_line": null, "item_type": "cython_function_or_method", "aliases": [ "numpy.random.standard_cauchy", "numpy.random.RandomState.standard_cauchy" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "Draw samples and plot the distribution:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import matplotlib.pyplot as plt\ns = np.random.standard_cauchy(1000000)\ns = s[(s>-25) & (s<25)] # truncate distribution so it plots well\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "plt.hist(s, bins=100)\n", "execution_status": "failure" }, { "__type": "Code", "__tag": 4050, "value": "plt.show()\n", "execution_status": "success" }, { "__type": "Figure", "__tag": 4024, "value": { "__type": "RefInfo", "__tag": 4000, "module": "numpy", "version": "2.4.4", "kind": "assets", "path": "fig-a6a1a5233e29f1b3.png" } } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "random.Generator.standard_cauchy", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "random.Generator.standard_cauchy" }, "kind": "module" }, "descriptions": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "which should be used for new code." } ] } ], "type": null } ], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "size", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "standard_cauchy" }, "references": [ ".. [1] NIST/SEMATECH e-Handbook of Statistical Methods, \"Cauchy", " Distribution\",", " https://www.itl.nist.gov/div898/handbook/eda/section3/eda3663.htm", ".. [2] Weisstein, Eric W. \"Cauchy Distribution.\" From MathWorld--A", " Wolfram Web Resource.", " https://mathworld.wolfram.com/CauchyDistribution.html", ".. [3] Wikipedia, \"Cauchy distribution\"", " https://en.wikipedia.org/wiki/Cauchy_distribution" ], "qa": "numpy.random:RandomState.standard_cauchy", "arbitrary": [], "local_refs": [ "samples", "size" ] }