{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The probability mass function (PMF) for the binomial distribution is" } ] }, { "__type": "Math", "__tag": 4058, "value": "P(N) = \\binom{n}{N}p^N(1-p)^{n-N}," }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "n" }, { "__type": "Text", "__tag": 4046, "value": " is the number of trials, " }, { "__type": "InlineMath", "__tag": 4057, "value": "p" }, { "__type": "Text", "__tag": 4046, "value": " is the probability of success, and " }, { "__type": "InlineMath", "__tag": 4057, "value": "N" }, { "__type": "Text", "__tag": 4046, "value": " is the number of successes." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "When estimating the standard error of a proportion in a population by using a random sample, the normal distribution works well unless the product p*n <=5, where p = population proportion estimate, and n = number of samples, in which case the binomial distribution is used instead. 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A company drills 9 wild-cat oil exploration\nwells, each with an estimated probability of success of 0.1. All nine\nwells fail. What is the probability of that happening?\n\nLet's do 20,000 trials of the model, and count the number that\ngenerate zero positive results.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "sum(np.random.binomial(9, 0.1, 20000) == 0)/20000.\n", "execution_status": "failure" } ], "title": [], "level": 0, "target": null }, "see_also": [ { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "random.Generator.binomial", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "random.Generator.binomial" }, "kind": "module" }, "descriptions": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "which should be used for new code." } ] } ], "type": null }, { "__type": "SeeAlsoItem", "__tag": 4028, "name": { "__type": "CrossRef", "__tag": 4002, "value": "scipy.stats.binom", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "current-module", "version": "current-version", "kind": "to-resolve", "path": "scipy.stats.binom" }, "kind": "module" }, "descriptions": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "probability density function, distribution or cumulative density function, etc." } ] } ], "type": null } ], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "n", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "p", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__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": "binomial" }, "references": [ ".. [1] Dalgaard, Peter, \"Introductory Statistics with R\",", " Springer-Verlag, 2002.", ".. [2] Glantz, Stanton A. \"Primer of Biostatistics.\", McGraw-Hill,", " Fifth Edition, 2002.", ".. [3] Lentner, Marvin, \"Elementary Applied Statistics\", Bogden", " and Quigley, 1972.", ".. [4] Weisstein, Eric W. \"Binomial Distribution.\" From MathWorld--A", " Wolfram Web Resource.", " https://mathworld.wolfram.com/BinomialDistribution.html", ".. [5] Wikipedia, \"Binomial distribution\",", " https://en.wikipedia.org/wiki/Binomial_distribution" ], "qa": "numpy.random:RandomState.binomial", "arbitrary": [], "local_refs": [ "n", "out", "p", "size" ] }