Revision #1430 → #1980 · back to history
modifiedNormal distribution20774e57c3c1
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| anchors | [{"section":"(Lead)","snippet":"a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable"},{"type":"math_alttext","value":"{\\displaystyle f(x)={\\frac {1}{\\sqrt {2\\pi \\sigma ^{2}}}}\\exp {\\left(-{\\frac {(x-\\mu )^{2}}{2\\sigma ^{2}}}\\right)}\\,.}"}] | — |
modifiedStandard normal distribution1209f7fb0984
| Field | From #1430 | To #1980 |
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| anchors | [{"section":"Standard normal distribution","snippet":"The simplest case of a normal distribution is known as the standard normal distribution or unit normal distribution"},{"type":"math_alttext","value":"{\\displaystyle \\varphi (z)={\\frac {e^{-z^{2}/2}}{\\sqrt {2\\pi }}}\\,.}"}] | — |
modifiedPrecision parameterizationdee38998bc1f
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Alternative parameterizations","snippet":"Some authors advocate using the precision"},{"type":"math_alttext","value":"{\\displaystyle f(x)={\\sqrt {\\frac {\\tau }{2\\pi }}}e^{-\\tau (x-\\mu )^{2}/2}.}"}] | — |
modifiedAlternative precision (reciprocal of standard deviation)2e7d5f6593ee
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Alternative parameterizations","snippet":"Alternatively, the reciprocal of the standard deviation"},{"type":"math_alttext","value":"{\\displaystyle f(x)={\\frac {\\tau '}{\\sqrt {2\\pi }}}e^{-(\\tau ')^{2}(x-\\mu )^{2}/2}.}"}] | — |
modifiedStandard normal CDF6dec3e90f695
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Cumulative distribution function","snippet":"The cumulative distribution function (CDF) of the standard normal distribution"},{"type":"math_alttext","value":"{\\displaystyle \\Phi (x)={\\frac {1}{\\sqrt {2\\pi }}}\\int _{-\\infty }^{x}e^{-t^{2}/2}\\,dt\\,.}"}] | — |
modifiedError function021da2ebe2ee
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Cumulative distribution function","snippet":"The related error function"},{"type":"math_alttext","value":"{\\displaystyle \\operatorname {erf} (x)={\\frac {1}{\\sqrt {\\pi }}}\\int _{-x}^{x}e^{-t^{2}}\\,dt={\\frac {2}{\\sqrt {\\pi }}}\\int _{0}^{x}e^{-t^{2}}\\,dt\\,.}"}] | — |
modifiedRelation between Phi and erf36e4357c7547
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Cumulative distribution function","snippet":"The two functions are closely related"},{"type":"math_alttext","value":"{\\displaystyle \\Phi (x)={\\frac {1}{2}}\\left[1+\\operatorname {erf} \\left({\\frac {x}{\\sqrt {2}}}\\right)\\right].}"}] | — |
modifiedGeneral normal CDFf94353e85339
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Cumulative distribution function","snippet":"For a generic normal distribution with density"},{"type":"math_alttext","value":"{\\displaystyle F(x)=\\Phi {\\left({\\frac {x-\\mu }{\\sigma }}\\right)}={\\frac {1}{2}}\\left[1+\\operatorname {erf} \\left({\\frac {x-\\mu }{\\sigma {\\sqrt {2}}}}\\right)\\right].}"}] | — |
modifiedProbability of interval8dfebded381a
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Cumulative distribution function","snippet":"The probability that x lies between a and b"},{"type":"math_alttext","value":"{\\displaystyle \\operatorname {P} (a<x\\leq b)={\\frac {1}{2}}\\left[\\operatorname {erf} \\left({\\frac {b-\\mu }{\\sigma {\\sqrt {2}}}}\\right)-\\operatorname {erf} \\left({\\frac {a-\\mu }{\\sigma {\\sqrt {2}}}}\\right)\\right]}"}] | — |
modifiedRotational symmetry of Phi4978d8a3317f
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Cumulative distribution function","snippet":"has 2-fold rotational symmetry around the point (0,1/2)"},{"type":"math_alttext","value":"{\\displaystyle \\int \\Phi (x)\\,dx=x\\Phi (x)+\\varphi (x)+C.}"}] | — |
modifiedAsymptotic expansion of CDF687d2cdef260
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Cumulative distribution function","snippet":"An asymptotic expansion of the cumulative distribution function for large x can be derived using integration by parts"},{"type":"math_alttext","value":"{\\displaystyle \\Phi (x)={\\frac {1}{2}}+{\\frac {1}{\\sqrt {2\\pi }}}e^{-x^{2}/2}\\sum _{n=0}^{\\infty }{\\frac {1}{(2n+1)!!}}x^{2n+1}\\,.}"}] | — |
modifiedRecursive Taylor expansion via Hermite polynomials23147b045e53
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Recursive computation with Taylor series","snippet":"The recurrence relation for Hermite polynomials"},{"type":"math_alttext","value":"{\\displaystyle \\Phi (x)=\\sum _{n=0}^{\\infty }{\\frac {\\Phi ^{(n)}(x_{0})}{n!}}(x-x_{0})^{n}\\,,}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\Phi ^{(0)}(x_{0})&={\\frac {1}{\\sqrt {2\\pi }}}\\int _{-\\infty }^{x_{0}}e^{-t^{2}/2}\\,dt\\\\\\Phi ^{(1)}(x_{0})&={\\frac {1}{\\sqrt {2\\pi }}}e^{-x_{0}^{2}/2}\\\\\\Phi ^{(n)}(x_{0})&=-\\left(x_{0}\\Phi ^{(n-1)}(x_{0})+(n-2)\\Phi ^{(n-2)}(x_{0})\\right),&n\\geq 2\\,.\\end{aligned}}}"}] | — |
modifiedQuantile/probit function955c5e4f7d71
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Quantile function","snippet":"The quantile function of a distribution is the inverse of the cumulative distribution function"},{"type":"math_alttext","value":"{\\displaystyle \\Phi ^{-1}(p)={\\sqrt {2}}\\operatorname {erf} ^{-1}(2p-1),\\quad p\\in (0,1).}"},{"type":"math_alttext","value":"{\\displaystyle F^{-1}(p)=\\mu +\\sigma \\Phi ^{-1}(p)=\\mu +\\sigma {\\sqrt {2}}\\operatorname {erf} ^{-1}(2p-1),\\quad p\\in (0,1).}"}] | — |
modifiedNewton's method for quantilea1cc565c8b06
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Using root finding to compute the quantile function","snippet":"can be used with Newton's method"},{"type":"math_alttext","value":"{\\displaystyle x_{n}=x_{n-1}-{\\frac {\\Phi (x_{n-1})-q}{\\varphi (x_{n-1})}}\\,.}"}] | — |
modifiedCentral moments of normalab37b9999642
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Moments","snippet":"the non-central moments exist and are finite for any"},{"type":"math_alttext","value":"{\\displaystyle \\operatorname {E} \\left[(X-\\mu )^{p}\\right]={\\begin{cases}0&{\\text{if }}p{\\text{ is odd,}}\\\\\\sigma ^{p}(p-1)!!&{\\text{if }}p{\\text{ is even.}}\\end{cases}}}"}] | — |
modifiedConditional expectation on an interval (inverse Mills ratio)2dd06fbc1252
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Moments","snippet":"The expectation of conditioned on the event"},{"type":"math_alttext","value":"{\\displaystyle \\operatorname {E} \\left[X\\mid a<X<b\\right]=\\mu -\\sigma ^{2}{\\frac {f(b)-f(a)}{F(b)-F(a)}}\\,,}"}] | — |
modifiedCharacteristic functiona8ae6eb63104
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Fourier transform and characteristic function","snippet":"is closely connected to the characteristic function"},{"type":"math_alttext","value":"{\\displaystyle \\varphi _{X}(t)={\\hat {f}}(-t)\\,.}"}] | — |
modifiedCumulant generating function5b94777b6a1d
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Moment- and cumulant-generating functions","snippet":"The cumulant generating function is the logarithm of the moment generating function"},{"type":"math_alttext","value":"{\\displaystyle g(t)=\\ln M(t)=\\mu t+{\\tfrac {1}{2}}\\sigma ^{2}t^{2}\\,.}"}] | — |
modifiedZero-variance limit (Dirac delta)414d89be61a8
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Zero-variance limit","snippet":"In the limit when"},{"type":"math_alttext","value":"{\\displaystyle F(x)={\\begin{cases}0&{\\text{if }}x<\\mu \\\\1&{\\text{if }}x\\geq \\mu .\\end{cases}}}"}] | — |
modifiedMaximum entropy theorem for normal531703f9b1ca
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Maximum entropy","snippet":"Of all probability distributions over the reals with a specified finite mean"},{"type":"math_alttext","value":"{\\displaystyle H(X)=-\\int _{-\\infty }^{\\infty }f(x)\\ln f(x)\\,dx\\,,}"},{"type":"math_alttext","value":"{\\displaystyle L=-\\int _{-\\infty }^{\\infty }f(x)\\ln f(x)\\,dx-\\lambda _{0}\\left(1-\\int _{-\\infty }^{\\infty }f(x)\\,dx\\right)-\\lambda _{1}\\left(\\mu -\\int _{-\\infty }^{\\infty }f(x)x\\,dx\\right)-\\lambda _{2}\\left(\\sigma ^{2}-\\int _{-\\infty }^{\\infty }f(x)(x-\\mu )^{2}\\,dx\\right)\\,.}"}] | — |
modifiedKullback-Leibler divergence of normals37d02de1e96d
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Other properties","snippet":"The Kullback–Leibler divergence of one normal distribution"},{"type":"math_alttext","value":"{\\displaystyle D_{\\mathrm {KL} }(X_{1}\\parallel X_{2})={\\frac {(\\mu _{1}-\\mu _{2})^{2}}{2\\sigma _{2}^{2}}}+{\\frac {1}{2}}\\left({\\frac {\\sigma _{1}^{2}}{\\sigma _{2}^{2}}}-1-\\ln {\\frac {\\sigma _{1}^{2}}{\\sigma _{2}^{2}}}\\right)}"},{"type":"math_alttext","value":"{\\displaystyle H^{2}(X_{1},X_{2})=1-{\\sqrt {\\frac {2\\sigma _{1}\\sigma _{2}}{\\sigma _{1}^{2}+\\sigma _{2}^{2}}}}\\exp \\left(-{\\frac {1}{4}}{\\frac {(\\mu _{1}-\\mu _{2})^{2}}{\\sigma _{1}^{2}+\\sigma _{2}^{2}}}\\right)}"}] | — |
| mathlib.decl | ProbabilityTheory.klDivergence | InformationTheory.klDiv |
| note | KL divergence is defined generally but the closed-form formula for two normals is not stated. | KL divergence between measures is defined generally as `InformationTheory.klDiv`, but the closed-form formula for two normals is not stated. |
addedHellinger distance between normals8523bd4a487b
modifiedFisher information matrix2c2557361bc4
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Other properties","snippet":"The Fisher information matrix for a normal distribution"},{"type":"math_alttext","value":"{\\displaystyle {\\mathcal {I}}(\\mu ,\\sigma ^{2})={\\begin{pmatrix}{\\frac {1}{\\sigma ^{2}}}&0\\\\0&{\\frac {1}{2\\sigma ^{4}}}\\end{pmatrix}}}"}] | — |
modifiedConjugate prior for the mean5906b7add1ec
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Other properties","snippet":"The conjugate prior of the mean of a normal distribution is another normal distribution"},{"type":"math_alttext","value":"{\\displaystyle \\mu \\mid x_{1},\\ldots ,x_{n}\\sim {\\mathcal {N}}\\left({\\frac {{\\frac {\\sigma ^{2}}{n}}\\mu _{0}+\\sigma _{0}^{2}{\\bar {x}}}{{\\frac {\\sigma ^{2}}{n}}+\\sigma _{0}^{2}}},\\left({\\frac {n}{\\sigma ^{2}}}+{\\frac {1}{\\sigma _{0}^{2}}}\\right)^{-1}\\right)}"}] | — |
addedStatistical manifold of normals19ae89e119fe
modifiedCentral limit theorem45e60730d645
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Central limit theorem","snippet":"The central limit theorem states that under certain (fairly common) conditions"},{"type":"math_alttext","value":"{\\displaystyle Z={\\sqrt {n}}{\\biggl (}{\\frac {1}{n}}\\sum _{i=1}^{n}X_{i}{\\biggr )}}"}] | — |
modifiedStability of normal distributionae4bed9155e0
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Operations on a single normal variable","snippet":"It follows that the normal distribution is stable"},{"type":"math_alttext","value":"{\\displaystyle X_{3}={\\frac {aX_{1}+bX_{2}-(a+b)\\mu }{\\sqrt {a^{2}+b^{2}}}}+\\mu }"}] | — |
modifiedSum of squares: chi-squareda1d987b30b02
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Operations on multiple independent normal variables","snippet":"then the sum of their squares has the chi-squared distribution"},{"type":"math_alttext","value":"{\\displaystyle X_{1}^{2}+\\cdots +X_{n}^{2}\\sim \\chi _{n}^{2}.}"}] | — |
modifiedSample mean and variance independence; t-distribution4bae1bb0bc77
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Operations on multiple independent normal variables","snippet":"then their sample mean is independent from the sample standard deviation"},{"type":"math_alttext","value":"{\\displaystyle t={\\frac {{\\overline {X}}-\\mu }{S/{\\sqrt {n}}}}={\\frac {{\\frac {1}{n}}(X_{1}+\\cdots +X_{n})-\\mu }{\\sqrt {{\\frac {1}{n(n-1)}}\\left[(X_{1}-{\\overline {X}})^{2}+\\cdots +(X_{n}-{\\overline {X}})^{2}\\right]}}}\\sim t_{n-1}.}"}] | — |
modifiedF-distribution ratio2e4f2a28635a
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Operations on multiple independent normal variables","snippet":"then the ratio of their normalized sums of squares will have the F-distribution"},{"type":"math_alttext","value":"{\\displaystyle F={\\frac {\\left(X_{1}^{2}+X_{2}^{2}+\\cdots +X_{n}^{2}\\right)/n}{\\left(Y_{1}^{2}+Y_{2}^{2}+\\cdots +Y_{m}^{2}\\right)/m}}\\sim F_{n,m}.}"}] | — |
modifiedTwo-piece normal distributione41414feceae
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Extensions","snippet":"A random variable X has a two-piece normal distribution"},{"type":"math_alttext","value":"{\\displaystyle f_{X}(x)={\\begin{cases}N(\\mu ,\\sigma _{1}^{2}),&{\\text{ if }}x\\leq \\mu \\\\N(\\mu ,\\sigma _{2}^{2}),&{\\text{ if }}x\\geq \\mu \\end{cases}}}"}] | — |
addedPearson distribution and generalized normald0aeba7aedc4
modifiedMaximum likelihood estimatorscb79cc5c9232
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Estimation of parameters","snippet":"The standard approach to this problem is the maximum likelihood method"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\ln {\\mathcal {L}}(\\mu ,\\sigma ^{2})&=\\sum _{i=1}^{n}\\ln f(x_{i}\\mid \\mu ,\\sigma ^{2})\\\\&=-{\\frac {n}{2}}\\ln(2\\pi )-{\\frac {n}{2}}\\ln \\sigma ^{2}-{\\frac {1}{2\\sigma ^{2}}}\\sum _{i=1}^{n}(x_{i}-\\mu )^{2}.\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\hat {\\mu }}={\\overline {x}}\\equiv {\\frac {1}{n}}\\sum _{i=1}^{n}x_{i},\\qquad {\\hat {\\sigma }}^{2}={\\frac {1}{n}}\\sum _{i=1}^{n}(x_{i}-{\\overline {x}})^{2}.}"}] | — |
modifiedSample meanf14574784b8d
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Sample mean","snippet":"is called the sample mean , since it is the arithmetic mean of all observations"},{"type":"math_alttext","value":"{\\displaystyle {\\hat {\\mu }}\\sim {\\mathcal {N}}(\\mu ,\\sigma ^{2}/n).}"}] | — |
modifiedLehmann-Scheffé UMVU for sample mean4f25a1406b9c
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Sample mean","snippet":"by the Lehmann–Scheffé theorem"},{"type":"math_alttext","value":"{\\displaystyle {\\hat {\\mu }}\\sim {\\mathcal {N}}(\\mu ,\\sigma ^{2}/n).}"}] | — |
modifiedSample variance and Bessel's correction63a893ae97b8
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Sample variance","snippet":"is called the sample variance , since it is the variance of the sample"},{"type":"math_alttext","value":"{\\displaystyle s^{2}={\\frac {n}{n-1}}{\\hat {\\sigma }}^{2}={\\frac {1}{n-1}}\\sum _{i=1}^{n}(x_{i}-{\\overline {x}})^{2}.}"},{"type":"math_alttext","value":"{\\displaystyle s^{2}\\sim {\\frac {\\sigma ^{2}}{n-1}}\\cdot \\chi _{n-1}^{2},\\qquad {\\hat {\\sigma }}^{2}\\sim {\\frac {\\sigma ^{2}}{n}}\\cdot \\chi _{n-1}^{2}.}"}] | — |
modifiedCochran's theorem (normal sample mean/variance independence)a3a3472445c1
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Confidence intervals","snippet":"By Cochran's theorem , for normal distributions the sample mean"},{"type":"math_alttext","value":"{\\displaystyle t={\\frac {{\\hat {\\mu }}-\\mu }{s/{\\sqrt {n}}}}={\\frac {{\\overline {x}}-\\mu }{\\sqrt {{\\frac {1}{n(n-1)}}\\sum (x_{i}-{\\overline {x}})^{2}}}}\\sim t_{n-1}}"},{"type":"math_alttext","value":"{\\displaystyle \\mu \\in \\left[{\\hat {\\mu }}-t_{n-1,1-\\alpha /2}{\\frac {s}{\\sqrt {n}}},\\,{\\hat {\\mu }}+t_{n-1,1-\\alpha /2}{\\frac {s}{\\sqrt {n}}}\\right]}"},{"type":"math_alttext","value":"{\\displaystyle \\sigma ^{2}\\in \\left[{\\frac {n-1}{\\chi _{n-1,1-\\alpha /2}^{2}}}s^{2},\\,{\\frac {n-1}{\\chi _{n-1,\\alpha /2}^{2}}}s^{2}\\right]}"}] | — |
modifiedConfidence intervals via t and chi-squarede5d71c2e6a80
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Confidence intervals","snippet":"This quantity t has the Student's t-distribution"},{"type":"math_alttext","value":"{\\displaystyle t={\\frac {{\\hat {\\mu }}-\\mu }{s/{\\sqrt {n}}}}={\\frac {{\\overline {x}}-\\mu }{\\sqrt {{\\frac {1}{n(n-1)}}\\sum (x_{i}-{\\overline {x}})^{2}}}}\\sim t_{n-1}}"},{"type":"math_alttext","value":"{\\displaystyle \\mu \\in \\left[{\\hat {\\mu }}-t_{n-1,1-\\alpha /2}{\\frac {s}{\\sqrt {n}}},\\,{\\hat {\\mu }}+t_{n-1,1-\\alpha /2}{\\frac {s}{\\sqrt {n}}}\\right]}"},{"type":"math_alttext","value":"{\\displaystyle \\sigma ^{2}\\in \\left[{\\frac {n-1}{\\chi _{n-1,1-\\alpha /2}^{2}}}s^{2},\\,{\\frac {n-1}{\\chi _{n-1,\\alpha /2}^{2}}}s^{2}\\right]}"}] | — |
modifiedBayesian update with known mean2fdeeee38527
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"With known mean","snippet":"the conjugate prior of the variance has an inverse gamma distribution"},{"type":"math_alttext","value":"{\\displaystyle p(\\sigma ^{2}\\mid \\nu _{0},\\sigma _{0}^{2})={\\frac {(\\sigma _{0}^{2}{\\frac {\\nu _{0}}{2}})^{\\nu _{0}/2}}{\\Gamma \\left({\\frac {\\nu _{0}}{2}}\\right)}}~{\\frac {\\exp \\left[{\\frac {-\\nu _{0}\\sigma _{0}^{2}}{2\\sigma ^{2}}}\\right]}{(\\sigma ^{2})^{1+{\\frac {\\nu _{0}}{2}}}}}\\propto {\\frac {\\exp \\left[{\\frac {-\\nu _{0}\\sigma _{0}^{2}}{2\\sigma ^{2}}}\\right]}{(\\sigma ^{2})^{1+{\\frac {\\nu _{0}}{2}}}}}}"}] | — |
modifiedBox-Muller methodb74241bf5ca7
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Generating values from normal distribution","snippet":"The Box–Muller method uses two independent random numbers"},{"type":"math_alttext","value":"{\\displaystyle X={\\sqrt {-2\\ln U}}\\,\\cos(2\\pi V),\\qquad Y={\\sqrt {-2\\ln U}}\\,\\sin(2\\pi V).}"}] | — |
modifiedMarsaglia polar method3c388bc88f87
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Generating values from normal distribution","snippet":"The Marsaglia polar method is a modification of the Box–Muller method"},{"type":"math_alttext","value":"{\\displaystyle X=U{\\sqrt {\\frac {-2\\ln S}{S}}},\\qquad Y=V{\\sqrt {\\frac {-2\\ln S}{S}}}}"}] | — |
modifiedZelen & Severo approximation851abb093c4d
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Numerical approximations for the normal cumulative distribution function and normal quantile function","snippet":"give the approximation for"},{"type":"math_alttext","value":"{\\displaystyle \\Phi (x)=1-\\varphi (x)\\left(b_{1}t+b_{2}t^{2}+b_{3}t^{3}+b_{4}t^{4}+b_{5}t^{5}\\right)+\\varepsilon (x),\\qquad t={\\frac {1}{1+b_{0}x}},}"}] | — |
modifiedMarsaglia Taylor-series algorithm3fdd36803276
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Numerical approximations for the normal cumulative distribution function and normal quantile function","snippet":"suggested a simple algorithm"},{"type":"math_alttext","value":"{\\displaystyle \\Phi (x)={\\frac {1}{2}}+\\varphi (x)\\left(x+{\\frac {x^{3}}{3}}+{\\frac {x^{5}}{3\\cdot 5}}+{\\frac {x^{7}}{3\\cdot 5\\cdot 7}}+{\\frac {x^{9}}{3\\cdot 5\\cdot 7\\cdot 9}}+\\cdots \\right)}"}] | — |
modifiedShore's approximationfc20abc5beeb
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Numerical approximations for the normal cumulative distribution function and normal quantile function","snippet":"introduced simple approximations that may be incorporated in stochastic optimization models"},{"type":"math_alttext","value":"{\\displaystyle z=\\Phi ^{-1}(p)=5.5556\\left[1-\\left({\\frac {1-p}{p}}\\right)^{0.1186}\\right],\\qquad p\\geq 1/2}"}] | — |
modifiedGauss's normal law of errorsea0970373945
| Field | From #1430 | To #1980 |
|---|
| anchors | [{"section":"Development","snippet":"Gauss demonstrates that the only law that rationalizes the choice of arithmetic mean as an estimator of the location parameter"},{"type":"math_alttext","value":"{\\displaystyle \\varphi {\\mathit {\\Delta }}={\\frac {h}{\\surd \\pi }}\\,e^{-\\mathrm {hh} \\Delta \\Delta },}"}] | — |