{ "__type": "IngestedDoc", "__tag": 4010, "_content": {}, "_ordered_sections": [], "item_file": null, "item_line": null, "item_type": null, "aliases": [], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "see_also": [], "signature": null, "references": null, "qa": "tutorial:stats:discrete", "arbitrary": [ { "__type": "Section", "__tag": 4015, "children": [], "title": [ { "__type": "Text", "__tag": 4046, "value": "Discrete Statistical Distributions" } ], "level": 0, "target": "discrete-random-variables" }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Discrete random variables take on only a countable number of values. The commonly used distributions are included in SciPy and described in this document. Each discrete distribution can take one extra integer parameter: " }, { "__type": "InlineMath", "__tag": 4057, "value": "L." }, { "__type": "Text", "__tag": 4046, "value": " The relationship between the general distribution " }, { "__type": "InlineMath", "__tag": 4057, "value": "p" }, { "__type": "Text", "__tag": 4046, "value": " and the standard distribution " }, { "__type": "InlineMath", "__tag": 4057, "value": "p_{0}" }, { "__type": "Text", "__tag": 4046, "value": " is" } ] }, { "__type": "Math", "__tag": 4058, "value": "p\\left(x\\right) = p_{0}\\left(x-L\\right)" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "which allows for shifting of the input. When a distribution generator is initialized, the discrete distribution can either specify the beginning and ending (integer) values " }, { "__type": "InlineMath", "__tag": 4057, "value": "a" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "b" }, { "__type": "Text", "__tag": 4046, "value": " which must be such that" } ] }, { "__type": "Math", "__tag": 4058, "value": "p_{0}\\left(x\\right) = 0\\quad x < a \\textrm{ or } x > b" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "in which case, it is assumed that the pdf function is specified on the integers " }, { "__type": "InlineMath", "__tag": 4057, "value": "a+mk\\leq b" }, { "__type": "Text", "__tag": 4046, "value": " where " }, { "__type": "InlineMath", "__tag": 4057, "value": "k" }, { "__type": "Text", "__tag": 4046, "value": " is a non-negative integer ( " }, { "__type": "InlineMath", "__tag": 4057, "value": "0,1,2,\\ldots" }, { "__type": "Text", "__tag": 4046, "value": " ) and " }, { "__type": "InlineMath", "__tag": 4057, "value": "m" }, { "__type": "Text", "__tag": 4046, "value": " is a positive integer multiplier. Alternatively, the two lists " }, { "__type": "InlineMath", "__tag": 4057, "value": "x_{k}" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "p\\left(x_{k}\\right)" }, { "__type": "Text", "__tag": 4046, "value": " can be provided directly in which case a dictionary is set up internally to evaluate probabilities and generate random variates." } ] } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Overview" } ], "level": 1, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The probability mass function of a random variable X is defined as the probability that the random variable takes on a particular value." } ] }, { "__type": "Math", "__tag": 4058, "value": "p\\left(x_{k}\\right)=P\\left[X=x_{k}\\right]" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "This is also sometimes called the probability density function, although technically" } ] }, { "__type": "Math", "__tag": 4058, "value": "f\\left(x\\right)=\\sum_{k}p\\left(x_{k}\\right)\\delta\\left(x-x_{k}\\right)" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "is the probability density function for a discrete distribution " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "1" }, { "__type": "Text", "__tag": 4046, "value": " ." } ] }, { "__type": "Footnote", "__tag": 4067, "label": "1", "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Note that we will be using " }, { "__type": "InlineMath", "__tag": 4057, "value": "p" }, { "__type": "Text", "__tag": 4046, "value": " to represent the probability mass function and a parameter (a probability). The usage should be obvious from context." } ] } ] } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Probability Mass Function (PMF)" } ], "level": 2, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The cumulative distribution function is" } ] }, { "__type": "Math", "__tag": 4058, "value": "F\\left(x\\right)=P\\left[X\\leq x\\right]=\\sum_{x_{k}\\leq x}p\\left(x_{k}\\right)" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "and is also useful to be able to compute. Note that" } ] }, { "__type": "Math", "__tag": 4058, "value": "F\\left(x_{k}\\right)-F\\left(x_{k-1}\\right)=p\\left(x_{k}\\right)" } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Cumulative Distribution Function (CDF)" } ], "level": 2, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The survival function is just" } ] }, { "__type": "Math", "__tag": 4058, "value": "S\\left(x\\right)=1-F\\left(x\\right)=P\\left[X>k\\right]" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "the probability that the random variable is strictly larger than " }, { "__type": "InlineMath", "__tag": 4057, "value": "k" }, { "__type": "Text", "__tag": 4046, "value": " ." } ] } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Survival Function" } ], "level": 2, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The percent point function is the inverse of the cumulative distribution function and is" } ] }, { "__type": "Math", "__tag": 4058, "value": "G\\left(q\\right)=F^{-1}\\left(q\\right)" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "for discrete distributions, this must be modified for cases where there is no " }, { "__type": "InlineMath", "__tag": 4057, "value": "x_{k}" }, { "__type": "Text", "__tag": 4046, "value": " such that " }, { "__type": "InlineMath", "__tag": 4057, "value": "F\\left(x_{k}\\right)=q." }, { "__type": "Text", "__tag": 4046, "value": " In these cases we choose " }, { "__type": "InlineMath", "__tag": 4057, "value": "G\\left(q\\right)" }, { "__type": "Text", "__tag": 4046, "value": " to be the smallest value " }, { "__type": "InlineMath", "__tag": 4057, "value": "x_{k}=G\\left(q\\right)" }, { "__type": "Text", "__tag": 4046, "value": " for which " }, { "__type": "InlineMath", "__tag": 4057, "value": "F\\left(x_{k}\\right)\\geq q" }, { "__type": "Text", "__tag": 4046, "value": " . If " }, { "__type": "InlineMath", "__tag": 4057, "value": "q=0" }, { "__type": "Text", "__tag": 4046, "value": " then we define " }, { "__type": "InlineMath", "__tag": 4057, "value": "G\\left(0\\right)=a-1" }, { "__type": "Text", "__tag": 4046, "value": " . This definition allows random variates to be defined in the same way as with continuous rv's using the inverse cdf on a uniform distribution to generate random variates." } ] } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Percent Point Function (Inverse CDF)" } ], "level": 2, "target": "discrete-ppf" }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The inverse survival function is the inverse of the survival function" } ] }, { "__type": "Math", "__tag": 4058, "value": "Z\\left(\\alpha\\right)=S^{-1}\\left(\\alpha\\right)=G\\left(1-\\alpha\\right)" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "and is thus the smallest non-negative integer " }, { "__type": "InlineMath", "__tag": 4057, "value": "k" }, { "__type": "Text", "__tag": 4046, "value": " for which " }, { "__type": "InlineMath", "__tag": 4057, "value": "F\\left(k\\right)\\geq1-\\alpha" }, { "__type": "Text", "__tag": 4046, "value": " or the smallest non-negative integer " }, { "__type": "InlineMath", "__tag": 4057, "value": "k" }, { "__type": "Text", "__tag": 4046, "value": " for which " }, { "__type": "InlineMath", "__tag": 4057, "value": "S\\left(k\\right)\\leq\\alpha." } ] } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Inverse survival function" } ], "level": 2, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If desired, the hazard function and the cumulative hazard function could be defined as" } ] }, { "__type": "Math", "__tag": 4058, "value": "h\\left(x_{k}\\right)=\\frac{p\\left(x_{k}\\right)}{1-F\\left(x_{k}\\right)}" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "and" } ] }, { "__type": "Math", "__tag": 4058, "value": "H\\left(x\\right)=\\sum_{x_{k}\\leq x}h\\left(x_{k}\\right)=\\sum_{x_{k}\\leq x}\\frac{F\\left(x_{k}\\right)-F\\left(x_{k-1}\\right)}{1-F\\left(x_{k}\\right)}." } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Hazard functions" } ], "level": 2, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Non-central moments are defined using the PDF" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\mu_{m}^{\\prime}=E\\left[X^{m}\\right]=\\sum_{k}x_{k}^{m}p\\left(x_{k}\\right)." }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Central moments are computed similarly " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\mu=\\mu_{1}^{\\prime}" } ] }, { "__type": "Math", "__tag": 4058, "value": "" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The mean is the first moment" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\mu=\\mu_{1}^{\\prime}=E\\left[X\\right]=\\sum_{k}x_{k}p\\left(x_{k}\\right)" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "the variance is the second central moment" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\mu_{2}=E\\left[\\left(X-\\mu\\right)^{2}\\right]=\\sum_{x_{k}}x_{k}^{2}p\\left(x_{k}\\right)-\\mu^{2}." }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Skewness is defined as" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\gamma_{1}=\\frac{\\mu_{3}}{\\mu_{2}^{3/2}}" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "while (Fisher) kurtosis is" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\gamma_{2}=\\frac{\\mu_{4}}{\\mu_{2}^{2}}-3," }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "so that a normal distribution has a kurtosis of zero." } ] } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Moments" } ], "level": 2, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The moment generating function is defined as" } ] }, { "__type": "Math", "__tag": 4058, "value": "M_{X}\\left(t\\right)=E\\left[e^{Xt}\\right]=\\sum_{x_{k}}e^{x_{k}t}p\\left(x_{k}\\right)" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Moments are found as the derivatives of the moment generating function evaluated at " }, { "__type": "InlineMath", "__tag": 4057, "value": "0." } ] } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Moment generating function" } ], "level": 2, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "To fit data to a distribution, maximizing the likelihood function is common. Alternatively, some distributions have well-known minimum variance unbiased estimators. These will be chosen by default, but the likelihood function will always be available for minimizing." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If " }, { "__type": "InlineMath", "__tag": 4057, "value": "f_{i}\\left(k;\\boldsymbol{\\theta}\\right)" }, { "__type": "Text", "__tag": 4046, "value": " is the PDF of a random-variable where " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\boldsymbol{\\theta}" }, { "__type": "Text", "__tag": 4046, "value": " is a vector of parameters ( " }, { "__type": "Emphasis", "__tag": 4047, "children": [ { "__type": "Text", "__tag": 4046, "value": "e.g." } ] }, { "__type": "Text", "__tag": 4046, "value": " " }, { "__type": "InlineMath", "__tag": 4057, "value": "L" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "S" }, { "__type": "Text", "__tag": 4046, "value": " ), then for a collection of " }, { "__type": "InlineMath", "__tag": 4057, "value": "N" }, { "__type": "Text", "__tag": 4046, "value": " independent samples from this distribution, the joint distribution the random vector " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\mathbf{k}" }, { "__type": "Text", "__tag": 4046, "value": " is" } ] }, { "__type": "Math", "__tag": 4058, "value": "f\\left(\\mathbf{k};\\boldsymbol{\\theta}\\right)=\\prod_{i=1}^{N}f_{i}\\left(k_{i};\\boldsymbol{\\theta}\\right)." }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The maximum likelihood estimate of the parameters " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\boldsymbol{\\theta}" }, { "__type": "Text", "__tag": 4046, "value": " are the parameters which maximize this function with " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\mathbf{x}" }, { "__type": "Text", "__tag": 4046, "value": " fixed and given by the data:" } ] }, { "__type": "Math", "__tag": 4058, "value": "" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Where" } ] }, { "__type": "Math", "__tag": 4058, "value": "" } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Fitting data" } ], "level": 2, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "We will use" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\overline{y\\left(\\mathbf{x}\\right)}=\\frac{1}{N}\\sum_{i=1}^{N}y\\left(x_{i}\\right)" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "N" }, { "__type": "Text", "__tag": 4046, "value": " should be clear from context." } ] } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Standard notation for mean" } ], "level": 2, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Note that" } ] }, { "__type": "Math", "__tag": 4058, "value": "k!=k\\cdot\\left(k-1\\right)\\cdot\\left(k-2\\right)\\cdot\\cdots\\cdot1=\\Gamma\\left(k+1\\right)" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "and has special cases of" } ] }, { "__type": "Math", "__tag": 4058, "value": "" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "and" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\left(\\begin{array}{c} n\\\\ k\\end{array}\\right)=\\frac{n!}{\\left(n-k\\right)!k!}." }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If " }, { "__type": "InlineMath", "__tag": 4057, "value": "n<0" }, { "__type": "Text", "__tag": 4046, "value": " or " }, { "__type": "InlineMath", "__tag": 4057, "value": "k<0" }, { "__type": "Text", "__tag": 4046, "value": " or " }, { "__type": "InlineMath", "__tag": 4057, "value": "k>n" }, { "__type": "Text", "__tag": 4046, "value": " we define " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\left(\\begin{array}{c} n\\\\ k\\end{array}\\right)=0" } ] } ], "title": [ { "__type": "Text", "__tag": 4046, "value": "Combinations" } ], "level": 2, "target": null }, { "__type": "Section", "__tag": 4015, "children": [ { "__type": "BulletList", "__tag": 4053, "ordered": false, "start": 1, "children": [ { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "CrossRef", "__tag": 4002, "value": "Bernoulli Distribution", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "docs", "path": "tutorial:stats:discrete_bernoulli" }, "kind": "exists" } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "CrossRef", "__tag": 4002, "value": 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