From ce7e3367c0f9477773fe76dd0eca64dc6ad33c02 Mon Sep 17 00:00:00 2001 From: Philipp Burckhardt Date: Sat, 8 Jul 2023 17:42:37 -0400 Subject: [PATCH] docs: update equations --- .../@stdlib/math/base/special/acovercos/README.md | 2 +- .../@stdlib/math/base/special/acoversin/README.md | 2 +- .../@stdlib/math/base/special/ahavercos/README.md | 2 +- .../@stdlib/math/base/special/ahaversin/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/avercos/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/aversin/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/beta/README.md | 4 ++-- lib/node_modules/@stdlib/math/base/special/betaln/README.md | 4 ++-- lib/node_modules/@stdlib/math/base/special/ccis/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/cexp/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/covercos/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/coversin/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/erf/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/erfc/README.md | 4 ++-- lib/node_modules/@stdlib/math/base/special/erfcinv/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/erfcx/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/erfinv/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/expit/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/gammasgn/README.md | 2 +- .../@stdlib/math/base/special/hacovercos/README.md | 2 +- .../@stdlib/math/base/special/hacoversin/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/havercos/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/haversin/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/logit/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/ramp/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/rampf/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/rcbrt/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/rsqrt/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/rsqrtf/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/signum/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/signumf/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/sinc/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/spence/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/vercos/README.md | 2 +- lib/node_modules/@stdlib/math/base/special/versin/README.md | 2 +- lib/node_modules/@stdlib/math/strided/special/ramp/README.md | 2 +- lib/node_modules/@stdlib/math/strided/special/rsqrt/README.md | 2 +- .../@stdlib/simulate/iter/lanczos-pulse/README.md | 2 +- .../@stdlib/simulate/iter/periodic-sinc/README.md | 2 +- .../@stdlib/simulate/iter/sawtooth-wave/README.md | 2 +- .../@stdlib/stats/base/dists/arcsine/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/arcsine/median/README.md | 2 +- 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2 +- .../@stdlib/stats/base/dists/betaprime/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/betaprime/variance/README.md | 2 +- .../@stdlib/stats/base/dists/binomial/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/binomial/median/README.md | 2 +- .../@stdlib/stats/base/dists/binomial/mode/README.md | 2 +- .../@stdlib/stats/base/dists/binomial/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/binomial/variance/README.md | 2 +- .../@stdlib/stats/base/dists/cauchy/median/README.md | 2 +- .../@stdlib/stats/base/dists/cauchy/mode/README.md | 2 +- .../@stdlib/stats/base/dists/chi/kurtosis/README.md | 2 +- lib/node_modules/@stdlib/stats/base/dists/chi/mode/README.md | 2 +- .../@stdlib/stats/base/dists/chi/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/chi/variance/README.md | 2 +- .../@stdlib/stats/base/dists/chisquare/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/chisquare/mode/README.md | 2 +- .../@stdlib/stats/base/dists/chisquare/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/chisquare/variance/README.md | 2 +- .../@stdlib/stats/base/dists/cosine/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/cosine/median/README.md | 2 +- .../@stdlib/stats/base/dists/cosine/mode/README.md | 2 +- .../@stdlib/stats/base/dists/cosine/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/cosine/variance/README.md | 2 +- .../@stdlib/stats/base/dists/degenerate/median/README.md | 2 +- .../@stdlib/stats/base/dists/degenerate/mode/README.md | 2 +- .../@stdlib/stats/base/dists/degenerate/stdev/README.md | 2 +- .../@stdlib/stats/base/dists/degenerate/variance/README.md | 2 +- .../stats/base/dists/discrete-uniform/kurtosis/README.md | 2 +- .../stats/base/dists/discrete-uniform/median/README.md | 2 +- .../stats/base/dists/discrete-uniform/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/discrete-uniform/stdev/README.md | 2 +- .../stats/base/dists/discrete-uniform/variance/README.md | 2 +- .../@stdlib/stats/base/dists/erlang/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/erlang/mode/README.md | 2 +- .../@stdlib/stats/base/dists/erlang/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/erlang/variance/README.md | 2 +- .../@stdlib/stats/base/dists/exponential/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/exponential/median/README.md | 2 +- .../@stdlib/stats/base/dists/exponential/mode/README.md | 2 +- .../@stdlib/stats/base/dists/exponential/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/exponential/variance/README.md | 2 +- .../@stdlib/stats/base/dists/f/kurtosis/README.md | 2 +- lib/node_modules/@stdlib/stats/base/dists/f/mode/README.md | 2 +- .../@stdlib/stats/base/dists/f/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/f/variance/README.md | 2 +- .../@stdlib/stats/base/dists/frechet/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/frechet/median/README.md | 2 +- .../@stdlib/stats/base/dists/frechet/mode/README.md | 2 +- .../@stdlib/stats/base/dists/frechet/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/frechet/variance/README.md | 2 +- .../@stdlib/stats/base/dists/gamma/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/gamma/mode/README.md | 2 +- .../@stdlib/stats/base/dists/gamma/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/gamma/variance/README.md | 2 +- .../@stdlib/stats/base/dists/geometric/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/geometric/median/README.md | 2 +- .../@stdlib/stats/base/dists/geometric/mode/README.md | 2 +- .../@stdlib/stats/base/dists/geometric/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/geometric/variance/README.md | 2 +- .../@stdlib/stats/base/dists/gumbel/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/gumbel/median/README.md | 2 +- .../@stdlib/stats/base/dists/gumbel/mode/README.md | 2 +- .../@stdlib/stats/base/dists/gumbel/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/gumbel/variance/README.md | 2 +- .../stats/base/dists/hypergeometric/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/hypergeometric/mode/README.md | 2 +- .../stats/base/dists/hypergeometric/skewness/README.md | 2 +- .../stats/base/dists/hypergeometric/variance/README.md | 2 +- .../@stdlib/stats/base/dists/invgamma/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/invgamma/mode/README.md | 2 +- .../@stdlib/stats/base/dists/invgamma/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/invgamma/variance/README.md | 2 +- .../@stdlib/stats/base/dists/kumaraswamy/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/kumaraswamy/median/README.md | 2 +- .../@stdlib/stats/base/dists/kumaraswamy/mode/README.md | 2 +- .../@stdlib/stats/base/dists/kumaraswamy/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/kumaraswamy/variance/README.md | 2 +- .../@stdlib/stats/base/dists/laplace/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/laplace/median/README.md | 2 +- 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2 +- .../stats/base/dists/negative-binomial/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/negative-binomial/mode/README.md | 2 +- .../stats/base/dists/negative-binomial/skewness/README.md | 2 +- .../stats/base/dists/negative-binomial/variance/README.md | 2 +- .../@stdlib/stats/base/dists/normal/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/normal/median/README.md | 2 +- .../@stdlib/stats/base/dists/normal/mode/README.md | 2 +- .../@stdlib/stats/base/dists/normal/quantile/README.md | 2 +- .../@stdlib/stats/base/dists/normal/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/normal/variance/README.md | 2 +- .../@stdlib/stats/base/dists/pareto-type1/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/pareto-type1/median/README.md | 2 +- .../@stdlib/stats/base/dists/pareto-type1/mode/README.md | 2 +- .../@stdlib/stats/base/dists/pareto-type1/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/pareto-type1/stdev/README.md | 2 +- .../@stdlib/stats/base/dists/pareto-type1/variance/README.md | 2 +- .../@stdlib/stats/base/dists/poisson/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/poisson/median/README.md | 2 +- .../@stdlib/stats/base/dists/poisson/mode/README.md | 2 +- .../@stdlib/stats/base/dists/poisson/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/poisson/variance/README.md | 2 +- .../@stdlib/stats/base/dists/rayleigh/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/rayleigh/median/README.md | 2 +- .../@stdlib/stats/base/dists/rayleigh/mode/README.md | 2 +- .../@stdlib/stats/base/dists/rayleigh/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/rayleigh/variance/README.md | 2 +- lib/node_modules/@stdlib/stats/base/dists/t/cdf/README.md | 2 +- .../@stdlib/stats/base/dists/t/kurtosis/README.md | 2 +- lib/node_modules/@stdlib/stats/base/dists/t/logcdf/README.md | 2 +- lib/node_modules/@stdlib/stats/base/dists/t/median/README.md | 2 +- lib/node_modules/@stdlib/stats/base/dists/t/mode/README.md | 2 +- .../@stdlib/stats/base/dists/t/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/t/variance/README.md | 2 +- .../@stdlib/stats/base/dists/triangular/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/triangular/median/README.md | 2 +- .../@stdlib/stats/base/dists/triangular/mode/README.md | 2 +- .../@stdlib/stats/base/dists/triangular/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/triangular/variance/README.md | 2 +- .../@stdlib/stats/base/dists/uniform/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/uniform/median/README.md | 2 +- .../@stdlib/stats/base/dists/uniform/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/uniform/variance/README.md | 2 +- .../@stdlib/stats/base/dists/weibull/kurtosis/README.md | 2 +- .../@stdlib/stats/base/dists/weibull/median/README.md | 2 +- .../@stdlib/stats/base/dists/weibull/mode/README.md | 2 +- .../@stdlib/stats/base/dists/weibull/skewness/README.md | 2 +- .../@stdlib/stats/base/dists/weibull/variance/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/apcorr/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/covariance/README.md | 4 ++-- lib/node_modules/@stdlib/stats/incr/covmat/README.md | 4 ++-- lib/node_modules/@stdlib/stats/incr/kurtosis/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/maape/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/mae/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/mapcorr/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/mape/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/mcovariance/README.md | 4 ++-- lib/node_modules/@stdlib/stats/incr/mda/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/me/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/mmaape/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/mmae/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/mmape/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/mmda/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/mme/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/mmpe/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/mmse/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/mpcorr/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/mpcorr2/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/mpcorrdist/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/mpe/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/mrmse/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/mrss/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/mse/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/rmse/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/rss/README.md | 2 +- lib/node_modules/@stdlib/stats/incr/skewness/README.md | 2 +- 227 files changed, 233 insertions(+), 233 deletions(-) diff --git a/lib/node_modules/@stdlib/math/base/special/acovercos/README.md b/lib/node_modules/@stdlib/math/base/special/acovercos/README.md index 581161e8826..a13c11d9a36 100644 --- a/lib/node_modules/@stdlib/math/base/special/acovercos/README.md +++ b/lib/node_modules/@stdlib/math/base/special/acovercos/README.md @@ -29,7 +29,7 @@ The [inverse coversed cosine][inverse-coversed-cosine] is defined as ```math -\operatorname{acovercos}(\theta) = \arcsin(1+\theta) +\mathop{\mathrm{acovercos}}(\theta) = \arcsin(1+\theta) ``` ```math -\operatorname{acoversin}(\theta) = \arcsin(1-\theta) +\mathop{\mathrm{acoversin}}(\theta) = \arcsin(1-\theta) ``` ```math -\operatorname{ahavercos}(\theta) = 2 \cdot \arccos(\sqrt{\theta}) +\mathop{\mathrm{ahavercos}}(\theta) = 2 \cdot \arccos(\sqrt{\theta}) ``` ```math -\operatorname{ahaversin}(\theta) = 2 \cdot \arcsin(\sqrt{\theta}) +\mathop{\mathrm{ahaversin}}(\theta) = 2 \cdot \arcsin(\sqrt{\theta}) ``` ```math -\operatorname{avercos}(\theta) = \arccos(1+\theta) +\mathop{\mathrm{avercos}}(\theta) = \arccos(1+\theta) ``` ```math -\operatorname{aversin}(\theta) = \arccos(1-\theta) +\mathop{\mathrm{aversin}}(\theta) = \arccos(1-\theta) ``` ```math -\operatorname{Beta}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,\mathrm{d}t +\mathop{\mathrm{Beta}}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,\mathrm{d}t ``` ```math -\operatorname{Beta}(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \! +\mathop{\mathrm{Beta}}(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \! ``` ```math -\operatorname{Beta}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,\mathrm{d}t +\mathop{\mathrm{Beta}}(x,y) = \int_0^1t^{x-1}(1-t)^{y-1}\,\mathrm{d}t ``` ```math -\operatorname{Beta}(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \! +\mathop{\mathrm{Beta}}(x,y)=\dfrac{\Gamma(x)\,\Gamma(y)}{\Gamma(x+y)} \! ``` ```math -\operatorname{cis}(z) = e^{iz} = \cos(z) + i \sin(z) +\mathop{\mathrm{cis}}(z) = e^{iz} = \cos(z) + i \sin(z) ``` ```math -\operatorname{exp}(z) = e^{x + i y} = (\exp{x}) (\cos(y) + i \sin(y)) +\mathop{\mathrm{exp}}(z) = e^{x + i y} = (\exp{x}) (\cos(y) + i \sin(y)) ``` ```math -\operatorname{covercos}(\theta) = 1 + \sin \theta +\mathop{\mathrm{covercos}}(\theta) = 1 + \sin \theta ``` ```math -\operatorname{coversin}(\theta) = 1 - \sin \theta +\mathop{\mathrm{coversin}}(\theta) = 1 - \sin \theta ``` ```math -\operatorname{erf}(x) = \frac{2}{\sqrt\pi}\int_0^x e^{-t^2}\,\mathrm dt +\mathop{\mathrm{erf}}(x) = \frac{2}{\sqrt\pi}\int_0^x e^{-t^2}\,\mathrm dt ``` ```math -\operatorname{erfc}(x) = 1 - \operatorname{erf}(x) = \frac{2}{\sqrt\pi} \int_x^{\infty} e^{-t^2}\, dt +\mathop{\mathrm{erfc}}(x) = 1 - \mathop{\mathrm{erf}}(x) = \frac{2}{\sqrt\pi} \int_x^{\infty} e^{-t^2}\, dt ``` ```math -\operatorname{erfc}(x) = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \exp \left( - \frac{x^2}{\sin^2 \theta} \right) d\theta +\mathop{\mathrm{erfc}}(x) = \frac{2}{\pi} \int_0^{\frac{\pi}{2}} \exp \left( - \frac{x^2}{\sin^2 \theta} \right) d\theta ``` ```math -\operatorname{erfc}^{-1}(1-z) = \operatorname{erf}^{-1}(z) +\mathop{\mathrm{erfc}}^{-1}(1-z) = \mathop{\mathrm{erf}}^{-1}(z) ``` ```math -\operatorname{erfcx}(x) = e^{x^2} \operatorname{erfc}(x) +\mathop{\mathrm{erfcx}}(x) = e^{x^2} \mathop{\mathrm{erfc}}(x) ``` ```math -\operatorname{erf}^{-1}(z)=\sum_{k=0}^\infty\frac{c_k}{2k+1}\left (\frac{\sqrt{\pi}}{2}z\right )^{2k+1} +\mathop{\mathrm{erf}}^{-1}(z)=\sum_{k=0}^\infty\frac{c_k}{2k+1}\left (\frac{\sqrt{\pi}}{2}z\right )^{2k+1} ``` ```math -\begin{aligned}\operatorname{expit}(x) &= \frac{1}{1+e^{-x}} \\ &= \frac{e^{x}}{e^{x}+1} \\ &= \frac{1}{2} + \frac{1}{2}\tanh\frac{x}{2} \end{aligned} +\begin{aligned}\mathop{\mathrm{expit}}(x) &= \frac{1}{1+e^{-x}} \\ &= \frac{e^{x}}{e^{x}+1} \\ &= \frac{1}{2} + \frac{1}{2}\tanh\frac{x}{2} \end{aligned} ``` ```math -\operatorname{gammasgn} ( x ) = \begin{cases} 1 & \textrm{if}\ \Gamma > 1 \\ -1 & \textrm{if}\ \Gamma < 1 \\ 0 & \textrm{otherwise}\ \end{cases} +\mathop{\mathrm{gammasgn}} ( x ) = \begin{cases} 1 & \textrm{if}\ \Gamma > 1 \\ -1 & \textrm{if}\ \Gamma < 1 \\ 0 & \textrm{otherwise}\ \end{cases} ``` ```math -\operatorname{hacovercos}(\theta) = \frac{1 + \sin \theta}{2} +\mathop{\mathrm{hacovercos}}(\theta) = \frac{1 + \sin \theta}{2} ``` ```math -\operatorname{hacoversin}(\theta) = \frac{1 - \sin \theta}{2} +\mathop{\mathrm{hacoversin}}(\theta) = \frac{1 - \sin \theta}{2} ``` ```math -\operatorname{havercos}(\theta) = \frac{1 + \cos \theta}{2} +\mathop{\mathrm{havercos}}(\theta) = \frac{1 + \cos \theta}{2} ``` ```math -\operatorname{haversin}(\theta) = \frac{1 - \cos \theta}{2} +\mathop{\mathrm{haversin}}(\theta) = \frac{1 - \cos \theta}{2} ``` ```math -\operatorname{logit}(p)=\log \left({\frac {p}{1-p}}\right) +\mathop{\mathrm{logit}}(p)=\log \left({\frac {p}{1-p}}\right) ``` ```math -R(x) = \operatorname{max}( x, 0 ) +R(x) = \mathop{\mathrm{max}}( x, 0 ) ``` ```math -R(x) = \operatorname{max}( x, 0 ) +R(x) = \mathop{\mathrm{max}}( x, 0 ) ``` ```math -\operatorname{rcbrt}(x)=\frac{1}{\sqrt[3]{x}} +\mathop{\mathrm{rcbrt}}(x)=\frac{1}{\sqrt[3]{x}} ``` ```math -\operatorname{rsqrt}(x)=\frac{1}{\sqrt{x}} +\mathop{\mathrm{rsqrt}}(x)=\frac{1}{\sqrt{x}} ``` ```math -\operatorname{rsqrtf}(x)=\frac{1}{\sqrt{x}} +\mathop{\mathrm{rsqrtf}}(x)=\frac{1}{\sqrt{x}} ``` ```math -\operatorname{sign}(x) := \begin{cases} -1 & \textrm{if}\ x < 0 \\ 0 & \textrm{if}\ x = 0 \\ 1 & \textrm{if}\ x > 0 \end{cases} +\mathop{\mathrm{sign}}(x) := \begin{cases} -1 & \textrm{if}\ x < 0 \\ 0 & \textrm{if}\ x = 0 \\ 1 & \textrm{if}\ x > 0 \end{cases} ``` ```math -\operatorname{sign}(x) := \begin{cases} -1 & \textrm{if}\ x < 0 \\ 0 & \textrm{if}\ x = 0 \\ 1 & \textrm{if}\ x > 0 \end{cases} +\mathop{\mathrm{sign}}(x) := \begin{cases} -1 & \textrm{if}\ x < 0 \\ 0 & \textrm{if}\ x = 0 \\ 1 & \textrm{if}\ x > 0 \end{cases} ``` ```math -\operatorname{sinc}(x) := \begin{cases} \frac {\sin(\pi x)}{\pi x} & \textrm{if}\ x \neq 0 \\ 1 & \textrm{if}\ x = 0 \end{cases} +\mathop{\mathrm{sinc}}(x) := \begin{cases} \frac {\sin(\pi x)}{\pi x} & \textrm{if}\ x \neq 0 \\ 1 & \textrm{if}\ x = 0 \end{cases} ``` ```math -\operatorname{Li}_{2}(z) = -\int_{0}^{z}{\ln(1-u) \over u}\,du{\text{, }}z\in \mathbb {C} +\mathop{\mathrm{Li}}_{2}(z) = -\int_{0}^{z}{\ln(1-u) \over u}\,du{\text{, }}z\in \mathbb {C} ``` ```math -\operatorname{vercos}(\theta) = 1 + \cos \theta +\mathop{\mathrm{vercos}}(\theta) = 1 + \cos \theta ``` ```math -\operatorname{versin}(\theta) = 1 - \cos \theta +\mathop{\mathrm{versin}}(\theta) = 1 - \cos \theta ``` ```math -R(x) = \operatorname{max}( x, 0 ) +R(x) = \mathop{\mathrm{max}}( x, 0 ) ``` ```math -\operatorname{rsqrt}(x)=\frac{1}{\sqrt{x}} +\mathop{\mathrm{rsqrt}}(x)=\frac{1}{\sqrt{x}} ``` ```math -f(t; T, \tau, a, \varphi) = \begin{cases}\operatorname{sinc}\biggl( \frac{2(t-\varphi)}{\tau-1} - 1\biggr) & (t-\varphi) \mod T < \tau \\ 0 & \textrm{otherwise} \end{cases} +f(t; T, \tau, a, \varphi) = \begin{cases}\mathop{\mathrm{sinc}}\biggl( \frac{2(t-\varphi)}{\tau-1} - 1\biggr) & (t-\varphi) \mod T < \tau \\ 0 & \textrm{otherwise} \end{cases} ``` ```math -D_N(\pi x; N, A) = A \cdot \frac{\operatorname{sinc}(Nx/2)} {\operatorname{sinc}(x/2)} +D_N(\pi x; N, A) = A \cdot \frac{\mathop{\mathrm{sinc}}(Nx/2)} {\mathop{\mathrm{sinc}}(x/2)} ``` ```math -f(t; \tau, a, \varphi) = \frac{2a}{\pi} \operatorname{arctan} \tan \frac{\pi(t-\varphi)}{\tau} +f(t; \tau, a, \varphi) = \frac{2a}{\pi} \mathop{\mathrm{arctan}} \tan \frac{\pi(t-\varphi)}{\tau} ``` ```math -\operatorname{Kurt}\left( X \right) = -{\frac {3}{5}} +\mathop{\mathrm{Kurt}}\left( X \right) = -{\frac {3}{5}} ``` ```math -\operatorname{Median}\left( X \right) = \frac{1}{2} \cdot ( a + b ) +\mathop{\mathrm{Median}}\left( X \right) = \frac{1}{2} \cdot ( a + b ) ``` ```math -\operatorname{mode}\left( X \right) = \{ a, b \} +\mathop{\mathrm{mode}}\left( X \right) = \{ a, b \} ``` ```math -\operatorname{skew}\left( X \right) = 0 +\mathop{\mathrm{skew}}\left( X \right) = 0 ``` ```math -\operatorname{Var}\left( X \right) = {\tfrac {1}{8}}(b-a)^{2} +\mathop{\mathrm{Var}}\left( X \right) = {\tfrac {1}{8}}(b-a)^{2} ``` ```math -\operatorname{Kurt}\left( X \right) = \frac{1}{pq} - 6 +\mathop{\mathrm{Kurt}}\left( X \right) = \frac{1}{pq} - 6 ``` ```math -\operatorname{Median}\left( X \right) = \begin{cases} 0 & \text{if } p \le 0.5 \\ 1 & \text{if } p > 0.5 \end{cases} +\mathop{\mathrm{Median}}\left( X \right) = \begin{cases} 0 & \text{if } p \le 0.5 \\ 1 & \text{if } p > 0.5 \end{cases} ``` ```math -\operatorname{Mode}\left( X \right) = \begin{cases} 0 & \text{if } p < 1/2 \\ 0, 1 &\text{if } p = 1/2 \\ 1 & \text{if } p > 1/2 \end{cases} +\mathop{\mathrm{Mode}}\left( X \right) = \begin{cases} 0 & \text{if } p < 1/2 \\ 0, 1 &\text{if } p = 1/2 \\ 1 & \text{if } p > 1/2 \end{cases} ``` ```math -\operatorname{skew}\left( X \right) = \frac{1-2p}{\sqrt{p(1-p)}} +\mathop{\mathrm{skew}}\left( X \right) = \frac{1-2p}{\sqrt{p(1-p)}} ``` ```math -\operatorname{Var}\left( X \right) = p \left( 1 - p \right) +\mathop{\mathrm{Var}}\left( X \right) = p \left( 1 - p \right) ``` ```math -F(x;\alpha,\beta) = \frac{\operatorname{Beta}(x;\alpha,\beta)}{\operatorname{Beta}(\alpha,\beta)} +F(x;\alpha,\beta) = \frac{\mathop{\mathrm{Beta}}(x;\alpha,\beta)}{\mathop{\mathrm{Beta}}(\alpha,\beta)} ``` ```math -\operatorname{Kurt}\left( X \right) = \frac{6[(\alpha - \beta)^2 (\alpha +\beta + 1) - \alpha \beta (\alpha + \beta + 2)]}{\alpha \beta (\alpha + \beta + 2) (\alpha + \beta + 3)} +\mathop{\mathrm{Kurt}}\left( X \right) = \frac{6[(\alpha - \beta)^2 (\alpha +\beta + 1) - \alpha \beta (\alpha + \beta + 2)]}{\alpha \beta (\alpha + \beta + 2) (\alpha + \beta + 3)} ``` ```math -\operatorname{Median}\left[ X \right] = I_{\frac{1}{2}}^{[-1]}(\alpha,\beta) +\mathop{\mathrm{Median}}\left[ X \right] = I_{\frac{1}{2}}^{[-1]}(\alpha,\beta) ``` ```math -\operatorname{mode}\left( X \right) = \frac{\alpha-1}{\alpha+\beta-2} +\mathop{\mathrm{mode}}\left( X \right) = \frac{\alpha-1}{\alpha+\beta-2} ``` ```math -\operatorname{skew}\left( X \right) = \frac{1-2p}{\sqrt{np(1-p)}} +\mathop{\mathrm{skew}}\left( X \right) = \frac{1-2p}{\sqrt{np(1-p)}} ``` ```math -\operatorname{Var}\left( X \right) = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)} +\mathop{\mathrm{Var}}\left( X \right) = \frac{\alpha\beta}{(\alpha+\beta)^2(\alpha+\beta+1)} ``` ```math -\operatorname{Kurt}\left( X \right) = 6{\frac{\alpha (\alpha +\beta -1)(5\beta -11)+(\beta -1)^{2}(\beta -2)}{\alpha (\alpha +\beta -1)(\beta -3)(\beta -4)}} +\mathop{\mathrm{Kurt}}\left( X \right) = 6{\frac{\alpha (\alpha +\beta -1)(5\beta -11)+(\beta -1)^{2}(\beta -2)}{\alpha (\alpha +\beta -1)(\beta -3)(\beta -4)}} ``` ```math -\operatorname{mode}(X) = \begin{cases}\frac{\alpha-1}{\beta+1} & \text{ if } \alpha \ge 1 \\ 0 & \text{ otherwise }\end{cases} +\mathop{\mathrm{mode}}(X) = \begin{cases}\frac{\alpha-1}{\beta+1} & \text{ if } \alpha \ge 1 \\ 0 & \text{ otherwise }\end{cases} ``` ```math -\operatorname{skew}\left( X \right) = \frac{2(2\alpha + \beta - 1)}{\beta - 3}{\sqrt{\frac{\beta - 2}{\alpha (\alpha + \beta - 1)}}} +\mathop{\mathrm{skew}}\left( X \right) = \frac{2(2\alpha + \beta - 1)}{\beta - 3}{\sqrt{\frac{\beta - 2}{\alpha (\alpha + \beta - 1)}}} ``` ```math -\operatorname{Var}\left( X \right) = \frac{\alpha(\alpha +\beta -1)}{(\beta-2)(\beta-1)^{2}} +\mathop{\mathrm{Var}}\left( X \right) = \frac{\alpha(\alpha +\beta -1)}{(\beta-2)(\beta-1)^{2}} ``` ```math -\operatorname{Kurt}\left( X \right) = {\frac {1-6p(1-p)}{np(1-p)}} +\mathop{\mathrm{Kurt}}\left( X \right) = {\frac {1-6p(1-p)}{np(1-p)}} ``` ```math -\operatorname{Median}\left( X \right) = [ n p ] +\mathop{\mathrm{Median}}\left( X \right) = [ n p ] ``` ```math -\operatorname{mode}\left( X \right) = \lfloor (n+1)p \rfloor +\mathop{\mathrm{mode}}\left( X \right) = \lfloor (n+1)p \rfloor ``` ```math -\operatorname{skew}\left( X \right) = \frac {1-2p}{\sqrt{np(1-p)}} +\mathop{\mathrm{skew}}\left( X \right) = \frac {1-2p}{\sqrt{np(1-p)}} ``` ```math -\operatorname{Var}\left[ X \right] = n p (1-p) +\mathop{\mathrm{Var}}\left[ X \right] = n p (1-p) ``` ```math -\operatorname{Median}\left( X \right) = x_0 +\mathop{\mathrm{Median}}\left( X \right) = x_0 ``` ```math -\operatorname{mode}\left( X \right) = x_0 +\mathop{\mathrm{mode}}\left( X \right) = x_0 ``` ```math -\operatorname{Kurt}\left( X \right) = {\frac{2}{\sigma ^{2}}}(1-\mu \sigma \gamma_{1}-\sigma^{2}) +\mathop{\mathrm{Kurt}}\left( X \right) = {\frac{2}{\sigma ^{2}}}(1-\mu \sigma \gamma_{1}-\sigma^{2}) ``` ```math -\operatorname{mode}\left( X \right) = \sqrt{k-1} +\mathop{\mathrm{mode}}\left( X \right) = \sqrt{k-1} ``` ```math -\operatorname{skew}\left( X \right) = \frac{\mu}{\sigma^{3}}\,(1 - 2 \sigma^{2}) +\mathop{\mathrm{skew}}\left( X \right) = \frac{\mu}{\sigma^{3}}\,(1 - 2 \sigma^{2}) ``` ```math -\operatorname{Var}\left( X \right) = k - \mathbb{E}\left[ X \right]^2 +\mathop{\mathrm{Var}}\left( X \right) = k - \mathbb{E}\left[ X \right]^2 ``` ```math -\operatorname{Kurt}\left( X \right) = \frac{12}{k} +\mathop{\mathrm{Kurt}}\left( X \right) = \frac{12}{k} ``` ```math -\operatorname{mode}\left( X \right) = \max(k-2,0) +\mathop{\mathrm{mode}}\left( X \right) = \max(k-2,0) ``` ```math -\operatorname{skew}\left( X \right) = \sqrt{8/k} +\mathop{\mathrm{skew}}\left( X \right) = \sqrt{8/k} ``` ```math -\operatorname{Var}\left( X \right) = 2k +\mathop{\mathrm{Var}}\left( X \right) = 2k ``` ```math -\operatorname{Kurt}\left( X \right) = \frac{6(90-\pi^{4})}{5(\pi^{2}-6)^{2}} +\mathop{\mathrm{Kurt}}\left( X \right) = \frac{6(90-\pi^{4})}{5(\pi^{2}-6)^{2}} ``` ```math -\operatorname{Median}\left( X \right) = \mu +\mathop{\mathrm{Median}}\left( X \right) = \mu ``` ```math -\operatorname{mode}\left( X \right) = \mu +\mathop{\mathrm{mode}}\left( X \right) = \mu ``` ```math -\operatorname{skew}\left( X \right) = 0 +\mathop{\mathrm{skew}}\left( X \right) = 0 ``` ```math -\operatorname{Var}\left( X \right) = s^{2}\left({\frac{1}{3}}-{\frac{2}{\pi^{2}}}\right) +\mathop{\mathrm{Var}}\left( X \right) = s^{2}\left({\frac{1}{3}}-{\frac{2}{\pi^{2}}}\right) ``` ```math -\operatorname{Median}\left( X \right) = \mu +\mathop{\mathrm{Median}}\left( X \right) = \mu ``` ```math -\operatorname{Mode}\left( X \right) = \mu +\mathop{\mathrm{Mode}}\left( X \right) = \mu ``` ```math -\operatorname{SD}\left( X \right) = 0 +\mathop{\mathrm{SD}}\left( X \right) = 0 ``` ```math -\operatorname{Var}\left( X \right) = 0 +\mathop{\mathrm{Var}}\left( X \right) = 0 ``` ```math -\operatorname{Kurt}\left( X \right) = -\frac{6(n^{2}+1)}{5(n^{2}-1)} +\mathop{\mathrm{Kurt}}\left( X \right) = -\frac{6(n^{2}+1)}{5(n^{2}-1)} ``` ```math -\operatorname{Median}\left[ X \right] = 0.5 \cdot ( a + b ) +\mathop{\mathrm{Median}}\left[ X \right] = 0.5 \cdot ( a + b ) ``` ```math -\operatorname{skew}\left( X \right) = 0 +\mathop{\mathrm{skew}}\left( X \right) = 0 ``` ```math -\operatorname{SD}\left( X \right) = \sqrt{ \frac{\left( b - a + 1 \right)^2 - 1}{12} } +\mathop{\mathrm{SD}}\left( X \right) = \sqrt{ \frac{\left( b - a + 1 \right)^2 - 1}{12} } ``` ```math -\operatorname{Var}\left( X \right) = \frac{\left( b - a + 1 \right)^2 - 1}{12} +\mathop{\mathrm{Var}}\left( X \right) = \frac{\left( b - a + 1 \right)^2 - 1}{12} ``` ```math -\operatorname{Kurt}\left( X \right) = \frac{6}{k} +\mathop{\mathrm{Kurt}}\left( X \right) = \frac{6}{k} ``` ```math -\operatorname{mode}\left( X \right) = \frac{1}{\lambda}(k - 1) +\mathop{\mathrm{mode}}\left( X \right) = \frac{1}{\lambda}(k - 1) ``` ```math -\operatorname{skew}\left( X \right) = \frac{2}{\sqrt{k}} +\mathop{\mathrm{skew}}\left( X \right) = \frac{2}{\sqrt{k}} ``` ```math -\operatorname{Var}\left( X \right) = \frac{k}{\lambda^{2}} +\mathop{\mathrm{Var}}\left( X \right) = \frac{k}{\lambda^{2}} ``` ```math -\operatorname{Kurt}\left( X \right) = 6 +\mathop{\mathrm{Kurt}}\left( X \right) = 6 ``` ```math -\operatorname{Median}\left( X \right) = \lambda^{−1} \ln(2) +\mathop{\mathrm{Median}}\left( X \right) = \lambda^{−1} \ln(2) ``` ```math -\operatorname{mode}\left( X \right) = 0 +\mathop{\mathrm{mode}}\left( X \right) = 0 ``` ```math -\operatorname{skew}\left( X \right) = 2 +\mathop{\mathrm{skew}}\left( X \right) = 2 ``` ```math -\operatorname{Var}\left( X \right) = \lambda^{-2} +\mathop{\mathrm{Var}}\left( X \right) = \lambda^{-2} ``` ```math -\operatorname{Kurt}\left( X \right) = \gamma_{2}=12{\frac{d_{1}(5d_{2}-22)(d_{1}+d_{2}-2)+(d_{2}-4)(d_{2}-2)^{2}}{d_{1}(d_{2}-6)(d_{2}-8)(d_{1}+d_{2}-2)}}. +\mathop{\mathrm{Kurt}}\left( X \right) = \gamma_{2}=12{\frac{d_{1}(5d_{2}-22)(d_{1}+d_{2}-2)+(d_{2}-4)(d_{2}-2)^{2}}{d_{1}(d_{2}-6)(d_{2}-8)(d_{1}+d_{2}-2)}}. ``` ```math -\operatorname{mode}\left( X \right) = \frac{d_{1}-2}{d_{1}} \; \frac{d_{2}}{d_{2}+2} +\mathop{\mathrm{mode}}\left( X \right) = \frac{d_{1}-2}{d_{1}} \; \frac{d_{2}}{d_{2}+2} ``` ```math -\operatorname{skew}\left( X \right) = \frac{(2d_{1}+d_{2}-2){\sqrt{8(d_{2}-4)}}}{(d_{2}-6){\sqrt{d_{1}(d_{1}+d_{2}-2)}}} +\mathop{\mathrm{skew}}\left( X \right) = \frac{(2d_{1}+d_{2}-2){\sqrt{8(d_{2}-4)}}}{(d_{2}-6){\sqrt{d_{1}(d_{1}+d_{2}-2)}}} ``` ```math -\operatorname{Var}\left( X \right) = \frac{2\,d_{2}^{2}\,(d_{1}+d_{2}-2)}{d_{1}(d_{2}-2)^{2}(d_{2}-4)} +\mathop{\mathrm{Var}}\left( X \right) = \frac{2\,d_{2}^{2}\,(d_{1}+d_{2}-2)}{d_{1}(d_{2}-2)^{2}(d_{2}-4)} ``` ```math -\operatorname{Kurt} = \begin{cases} -6+{\frac{\Gamma \left(1-{\frac{4}{\alpha }}\right)-4\Gamma \left(1-{\frac{3}{\alpha }}\right)\Gamma \left(1-{\frac{1}{\alpha }}\right)+3\Gamma^{2}\left(1-{\frac{2}{\alpha }}\right)}{\left[\Gamma \left(1-{\frac{2}{\alpha }}\right)-\Gamma^{2}\left(1-{\frac{1}{\alpha }}\right)\right]^{2}}} & {\text{ for }}\alpha >4\\\ \infty & \text{ otherwise }\end{cases} +\mathop{\mathrm{Kurt}} = \begin{cases} -6+{\frac{\Gamma \left(1-{\frac{4}{\alpha }}\right)-4\Gamma \left(1-{\frac{3}{\alpha }}\right)\Gamma \left(1-{\frac{1}{\alpha }}\right)+3\Gamma^{2}\left(1-{\frac{2}{\alpha }}\right)}{\left[\Gamma \left(1-{\frac{2}{\alpha }}\right)-\Gamma^{2}\left(1-{\frac{1}{\alpha }}\right)\right]^{2}}} & {\text{ for }}\alpha >4\\\ \infty & \text{ otherwise }\end{cases} ``` ```math -\operatorname{Median} = m+{\frac {s}{{\sqrt[ {\alpha }]{\ln(2)}}}} +\mathop{\mathrm{Median}} = m+{\frac {s}{{\sqrt[ {\alpha }]{\ln(2)}}}} ``` ```math -\operatorname{mode} = m+s\left({\frac{\alpha}{1+\alpha }}\right)^{{1/\alpha }} +\mathop{\mathrm{mode}} = m+s\left({\frac{\alpha}{1+\alpha }}\right)^{{1/\alpha }} ``` ```math -\operatorname{skew} = \begin{cases} {\frac{\Gamma \left(1-{\frac{3}{\alpha }}\right)-3\Gamma \left(1-{\frac{2}{\alpha }}\right)\Gamma \left(1-{\frac{1}{\alpha }}\right)+2\Gamma^{3}\left(1-{\frac{1}{\alpha }}\right)}{{\sqrt{\left(\Gamma \left(1-{\frac{2}{\alpha }}\right)-\Gamma^{2}\left(1-{\frac{1}{\alpha }}\right)\right)^{3}}}}} & \text{ for }\alpha > 3\\\ \infty & \text{ otherwise }\end{cases} +\mathop{\mathrm{skew}} = \begin{cases} {\frac{\Gamma \left(1-{\frac{3}{\alpha }}\right)-3\Gamma \left(1-{\frac{2}{\alpha }}\right)\Gamma \left(1-{\frac{1}{\alpha }}\right)+2\Gamma^{3}\left(1-{\frac{1}{\alpha }}\right)}{{\sqrt{\left(\Gamma \left(1-{\frac{2}{\alpha }}\right)-\Gamma^{2}\left(1-{\frac{1}{\alpha }}\right)\right)^{3}}}}} & \text{ for }\alpha > 3\\\ \infty & \text{ otherwise }\end{cases} ``` ```math -\operatorname{Var}\left( X \right) = \begin{cases} s^{2}\left(\Gamma \left(1-{\frac{2}{\alpha }}\right)-\left(\Gamma\left(1-{\frac {1}{\alpha }}\right)\right)^{2}\right) & {\text {for }}\alpha > 2\\\ \infty & \text{ otherwise } \end{cases} +\mathop{\mathrm{Var}}\left( X \right) = \begin{cases} s^{2}\left(\Gamma \left(1-{\frac{2}{\alpha }}\right)-\left(\Gamma\left(1-{\frac {1}{\alpha }}\right)\right)^{2}\right) & {\text {for }}\alpha > 2\\\ \infty & \text{ otherwise } \end{cases} ``` ```math -\operatorname{Kurt}\left( X \right) = \frac{6}{\alpha} +\mathop{\mathrm{Kurt}}\left( X \right) = \frac{6}{\alpha} ``` ```math -\operatorname{mode}\left( X \right) = \frac{\alpha \,-\,1}{\beta } +\mathop{\mathrm{mode}}\left( X \right) = \frac{\alpha \,-\,1}{\beta } ``` ```math -\operatorname{skew}\left( X \right) = \frac{2}{\sqrt{\alpha}} +\mathop{\mathrm{skew}}\left( X \right) = \frac{2}{\sqrt{\alpha}} ``` ```math -\operatorname{Var}\left( X \right) = \frac{\alpha}{\beta^2} +\mathop{\mathrm{Var}}\left( X \right) = \frac{\alpha}{\beta^2} ``` ```math -\operatorname{Kurt}\left( X \right) = 6+{\frac{p^{2}}{1-p}} +\mathop{\mathrm{Kurt}}\left( X \right) = 6+{\frac{p^{2}}{1-p}} ``` ```math -\operatorname{Median}\left( X \right) = \left\lceil {\frac {-1}{\log _{2}(1-p)}}\right\rceil \!-1 +\mathop{\mathrm{Median}}\left( X \right) = \left\lceil {\frac {-1}{\log _{2}(1-p)}}\right\rceil \!-1 ``` ```math -\operatorname{mode}\left( X \right) = 0 +\mathop{\mathrm{mode}}\left( X \right) = 0 ``` ```math -\operatorname{skew}\left( X \right) = \frac{2-p}{\sqrt{1-p}} +\mathop{\mathrm{skew}}\left( X \right) = \frac{2-p}{\sqrt{1-p}} ``` ```math -\operatorname{Var}\left( X \right) = \frac{1-p}{p^{2}} +\mathop{\mathrm{Var}}\left( X \right) = \frac{1-p}{p^{2}} ``` ```math -\operatorname{Kurt}\left( X \right) = \frac{12}{5} +\mathop{\mathrm{Kurt}}\left( X \right) = \frac{12}{5} ``` ```math -\operatorname{Median}\left( X \right) = \mu -\beta \,\ln(\ln(2)) +\mathop{\mathrm{Median}}\left( X \right) = \mu -\beta \,\ln(\ln(2)) ``` ```math -\operatorname{mode}\left( X \right) = \mu +\mathop{\mathrm{mode}}\left( X \right) = \mu ``` ```math -\operatorname{skew}\left( X \right) = {\frac{12{\sqrt{6}}\,\zeta(3)}{\pi^{3}}} \approx 1.14 +\mathop{\mathrm{skew}}\left( X \right) = {\frac{12{\sqrt{6}}\,\zeta(3)}{\pi^{3}}} \approx 1.14 ``` ```math -\operatorname{Var}\left( X \right) = \frac{\pi^{2}}{6}\,\beta^{2} +\mathop{\mathrm{Var}}\left( X \right) = \frac{\pi^{2}}{6}\,\beta^{2} ``` ```math -\operatorname{Kurt}\left( X \right) = \frac{(N-1) N^{2} \left[ N(N+1)-6K(N-K)-6n(N-n) \right]+6nK(N-K)(N-n)(5N-6)}{nK(N-K)(N-n)(N-2)(N-3)} +\mathop{\mathrm{Kurt}}\left( X \right) = \frac{(N-1) N^{2} \left[ N(N+1)-6K(N-K)-6n(N-n) \right]+6nK(N-K)(N-n)(5N-6)}{nK(N-K)(N-n)(N-2)(N-3)} ``` ```math -\operatorname{mode}\left( X \right) = \left\lfloor {\frac{(n+1)(K+1)}{N+2}}\right\rfloor +\mathop{\mathrm{mode}}\left( X \right) = \left\lfloor {\frac{(n+1)(K+1)}{N+2}}\right\rfloor ``` ```math -\operatorname{skew}\left( X \right) = \frac{(N-2K)(N-1)^{\frac{1}{2}}(N-2n)}{[nK(N-K)(N-n)]^{\frac{1}{2}}(N-2)} +\mathop{\mathrm{skew}}\left( X \right) = \frac{(N-2K)(N-1)^{\frac{1}{2}}(N-2n)}{[nK(N-K)(N-n)]^{\frac{1}{2}}(N-2)} ``` ```math -\operatorname{Var}\left( X \right) = n{K \over N}{(N-K) \over N}{N-n \over N-1} +\mathop{\mathrm{Var}}\left( X \right) = n{K \over N}{(N-K) \over N}{N-n \over N-1} ``` ```math -\operatorname{Kurt}\left( X \right) = \frac {30\,\alpha -66}{(\alpha -3)(\alpha -4)} +\mathop{\mathrm{Kurt}}\left( X \right) = \frac {30\,\alpha -66}{(\alpha -3)(\alpha -4)} ``` ```math -\operatorname{mode}\left( X \right) = \frac{\beta}{\alpha+1} +\mathop{\mathrm{mode}}\left( X \right) = \frac{\beta}{\alpha+1} ``` ```math -\operatorname{skew}\left( X \right) = \frac{4\sqrt{\alpha-2}}{\alpha-3} +\mathop{\mathrm{skew}}\left( X \right) = \frac{4\sqrt{\alpha-2}}{\alpha-3} ``` ```math -\operatorname{Var}\left( X \right) = \frac{\beta^{2}}{(\alpha-1)^{2}(\alpha-2)} +\mathop{\mathrm{Var}}\left( X \right) = \frac{\beta^{2}}{(\alpha-1)^{2}(\alpha-2)} ``` ```math -\operatorname{Kurt}\left( X \right) = \frac{ m_4 - ( 4.0 \cdot m_3 \cdot m_1 ) + ( 6.0 \cdot m_2 \cdot m_1^2 ) - ( 3.0 \cdot m_1^4 ) ) }{\left( m_2 - m_1^2 \right)^2} +\mathop{\mathrm{Kurt}}\left( X \right) = \frac{ m_4 - ( 4.0 \cdot m_3 \cdot m_1 ) + ( 6.0 \cdot m_2 \cdot m_1^2 ) - ( 3.0 \cdot m_1^4 ) ) }{\left( m_2 - m_1^2 \right)^2} ``` ```math -\operatorname{Median}\left( X \right) = \left(1-2^{{-1/b}}\right)^{1/a} +\mathop{\mathrm{Median}}\left( X \right) = \left(1-2^{{-1/b}}\right)^{1/a} ``` ```math -\operatorname{mode}\left( X \right) = \left(\frac{a-1}{ab-1}\right)^{1/a} +\mathop{\mathrm{mode}}\left( X \right) = \left(\frac{a-1}{ab-1}\right)^{1/a} ``` ```math -\operatorname{skew}\left( X \right) = \frac{ m_3 - 3m_1\sigma^2 - m_1^3 }{ \left( \sigma^2 \right)^{3/2} } +\mathop{\mathrm{skew}}\left( X \right) = \frac{ m_3 - 3m_1\sigma^2 - m_1^3 }{ \left( \sigma^2 \right)^{3/2} } ``` ```math -\operatorname{Var}\left( X \right) = m_2 - m_1^2 +\mathop{\mathrm{Var}}\left( X \right) = m_2 - m_1^2 ``` ```math -\operatorname{Kurt}\left( X \right) = 3 +\mathop{\mathrm{Kurt}}\left( X \right) = 3 ``` ```math -\operatorname{Median}\left( X \right) = \mu +\mathop{\mathrm{Median}}\left( X \right) = \mu ``` ```math -\operatorname{mode}\left( X \right) = \mu +\mathop{\mathrm{mode}}\left( X \right) = \mu ``` ```math -\operatorname{skew}\left( X \right) = 0 +\mathop{\mathrm{skew}}\left( X \right) = 0 ``` ```math -\operatorname{Var}\left( X \right) = 2 b^2 +\mathop{\mathrm{Var}}\left( X \right) = 2 b^2 ``` ```math -\operatorname{Median}\left( X \right) = \mu + \frac{c}{2(\mathop{\mathrm{erfcinv}}(1/2))^2} +\mathop{\mathrm{Median}}\left( X \right) = \mu + \frac{c}{2(\mathop{\mathrm{erfcinv}}(1/2))^2} ``` ```math -\operatorname{mode}\left( X \right) = \mu + \frac{c}{3} +\mathop{\mathrm{mode}}\left( X \right) = \mu + \frac{c}{3} ``` ```math -\operatorname{Var}\left( X \right) = \infty +\mathop{\mathrm{Var}}\left( X \right) = \infty ``` ```math -\operatorname{Kurt}\left( X \right) = 1.2 +\mathop{\mathrm{Kurt}}\left( X \right) = 1.2 ``` ```math -\operatorname{Median}\left( X \right) = \mu +\mathop{\mathrm{Median}}\left( X \right) = \mu ``` ```math -\operatorname{mode}\left( X \right) = \mu +\mathop{\mathrm{mode}}\left( X \right) = \mu ``` ```math -\operatorname{skew}\left( X \right) = 0 +\mathop{\mathrm{skew}}\left( X \right) = 0 ``` ```math -\operatorname{Var}\left( X \right) = \tfrac{s^{2}\pi^{2}}{3} +\mathop{\mathrm{Var}}\left( X \right) = \tfrac{s^{2}\pi^{2}}{3} ``` ```math -\operatorname{Kurt}\left( X \right) = \exp\left({4\sigma^{2}}\right)+2\exp\left({3\sigma^{2}}\right)+3\exp\left({2\sigma^{2}}\right)-6 +\mathop{\mathrm{Kurt}}\left( X \right) = \exp\left({4\sigma^{2}}\right)+2\exp\left({3\sigma^{2}}\right)+3\exp\left({2\sigma^{2}}\right)-6 ``` ```math -\operatorname{Median}\left( X \right) = \exp(\mu) +\mathop{\mathrm{Median}}\left( X \right) = \exp(\mu) ``` ```math -\operatorname{mode}\left( X \right) = \exp({\mu -\sigma^{2}}) +\mathop{\mathrm{mode}}\left( X \right) = \exp({\mu -\sigma^{2}}) ``` ```math -\operatorname{skew}\left( X \right) = \left(\exp(\sigma^{2}\right)\!\!+2){\sqrt{\exp(\sigma ^{2})\!\!-1}} +\mathop{\mathrm{skew}}\left( X \right) = \left(\exp(\sigma^{2}\right)\!\!+2){\sqrt{\exp(\sigma ^{2})\!\!-1}} ``` ```math -\operatorname{Var}\left( X \right) = [\exp({\sigma^{2}})-1] \cdot \exp({2\mu +\sigma^{2}}) +\mathop{\mathrm{Var}}\left( X \right) = [\exp({\sigma^{2}})-1] \cdot \exp({2\mu +\sigma^{2}}) ``` ```math -\operatorname{Kurt}\left( X \right) = \frac{6}{r} + \frac{(1-p)^{2}}{pr} +\mathop{\mathrm{Kurt}}\left( X \right) = \frac{6}{r} + \frac{(1-p)^{2}}{pr} ``` ```math -\operatorname{mode}\left( X \right) = \begin{cases}{\big \lfloor }{\frac{p(r-1)}{1-p}}{\big \rfloor } & \text{ if }\ r>1\\ 0 & \text{ if } \ r\leq 1\end{cases} +\mathop{\mathrm{mode}}\left( X \right) = \begin{cases}{\big \lfloor }{\frac{p(r-1)}{1-p}}{\big \rfloor } & \text{ if }\ r>1\\ 0 & \text{ if } \ r\leq 1\end{cases} ``` ```math -\operatorname{skew}\left( X \right) = \frac{1+p}{\sqrt{pr}} +\mathop{\mathrm{skew}}\left( X \right) = \frac{1+p}{\sqrt{pr}} ``` ```math -\operatorname{Var}\left( X \right) = \frac{pr}{(1-p)^{2}} +\mathop{\mathrm{Var}}\left( X \right) = \frac{pr}{(1-p)^{2}} ``` ```math -\operatorname{Kurt}\left( X \right) = 0 +\mathop{\mathrm{Kurt}}\left( X \right) = 0 ``` ```math -\operatorname{Median}\left( X \right) = \mu +\mathop{\mathrm{Median}}\left( X \right) = \mu ``` ```math -\operatorname{mode}\left( X \right) = \mu +\mathop{\mathrm{mode}}\left( X \right) = \mu ``` ```math -Q(p;\mu,\sigma) = \mu+\sigma\sqrt{2}\,\operatorname{erf}^{-1}(2p-1) +Q(p;\mu,\sigma) = \mu+\sigma\sqrt{2}\,\mathop{\mathrm{erf}}^{-1}(2p-1) ``` ```math -\operatorname{skew}\left( X \right) = 0 +\mathop{\mathrm{skew}}\left( X \right) = 0 ``` ```math -\operatorname{Var}\left[ X \right] = \sigma^2 +\mathop{\mathrm{Var}}\left[ X \right] = \sigma^2 ``` ```math -\operatorname{Kurt}\left( X \right) = \frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)} +\mathop{\mathrm{Kurt}}\left( X \right) = \frac{6(\alpha^3+\alpha^2-6\alpha-2)}{\alpha(\alpha-3)(\alpha-4)} ``` ```math -\operatorname{Median}\left( X \right) = \beta \sqrt[\alpha]{2} +\mathop{\mathrm{Median}}\left( X \right) = \beta \sqrt[\alpha]{2} ``` ```math -\operatorname{mode}\left( X \right) = \beta +\mathop{\mathrm{mode}}\left( X \right) = \beta ``` ```math -\operatorname{skew}\left( X \right) = \frac{2(1+\alpha)}{\alpha-3}\,\sqrt{\frac{\alpha-2}{\alpha}} +\mathop{\mathrm{skew}}\left( X \right) = \frac{2(1+\alpha)}{\alpha-3}\,\sqrt{\frac{\alpha-2}{\alpha}} ``` ```math -\operatorname{SD}\left( X \right) = \begin{cases} \infty & \text{for }\alpha\in(0,2] \\ \sqrt{ \frac{\beta^2\alpha}{(\alpha-1)^2(\alpha-2)} } & \text{for }\alpha>2 \end{cases} +\mathop{\mathrm{SD}}\left( X \right) = \begin{cases} \infty & \text{for }\alpha\in(0,2] \\ \sqrt{ \frac{\beta^2\alpha}{(\alpha-1)^2(\alpha-2)} } & \text{for }\alpha>2 \end{cases} ``` ```math -\operatorname{Var}\left( X \right) = \begin{cases} \infty & \text{for }\alpha\in(0,2] \\ \frac{\beta^2\alpha}{(\alpha-1)^2(\alpha-2)} & \text{for }\alpha>2 \end{cases} +\mathop{\mathrm{Var}}\left( X \right) = \begin{cases} \infty & \text{for }\alpha\in(0,2] \\ \frac{\beta^2\alpha}{(\alpha-1)^2(\alpha-2)} & \text{for }\alpha>2 \end{cases} ``` ```math -\operatorname{Kurt}\left( X \right) = \lambda^{-1} +\mathop{\mathrm{Kurt}}\left( X \right) = \lambda^{-1} ``` ```math -\operatorname{Median}\left( X \right) = \lfloor \lambda+1/3-0.02/\lambda \rfloor +\mathop{\mathrm{Median}}\left( X \right) = \lfloor \lambda+1/3-0.02/\lambda \rfloor ``` ```math -\operatorname{mode}\left( X \right) = \lfloor \lambda \rfloor +\mathop{\mathrm{mode}}\left( X \right) = \lfloor \lambda \rfloor ``` ```math -\operatorname{skew}\left( X \right) = \lambda^{-1/2} +\mathop{\mathrm{skew}}\left( X \right) = \lambda^{-1/2} ``` ```math -\operatorname{Var}\left( X \right) = \lambda +\mathop{\mathrm{Var}}\left( X \right) = \lambda ``` ```math -\operatorname{Kurt}\left( X \right) = \sigma \sqrt{2 \ln(2)} +\mathop{\mathrm{Kurt}}\left( X \right) = \sigma \sqrt{2 \ln(2)} ``` ```math -\operatorname{Median}\left( X \right) = \sigma \sqrt{2 \ln(2)} +\mathop{\mathrm{Median}}\left( X \right) = \sigma \sqrt{2 \ln(2)} ``` ```math -\operatorname{mode}\left( X \right) = \sigma +\mathop{\mathrm{mode}}\left( X \right) = \sigma ``` ```math -\operatorname{skew}\left( X \right) = \frac{2{\sqrt{\pi }}(\pi -3)}{(4-\pi )^{3/2}} +\mathop{\mathrm{skew}}\left( X \right) = \frac{2{\sqrt{\pi }}(\pi -3)}{(4-\pi )^{3/2}} ``` ```math -\operatorname{Var}\left( X \right) = \frac{4-\pi }{2}\sigma^{2} +\mathop{\mathrm{Var}}\left( X \right) = \frac{4-\pi }{2}\sigma^{2} ``` ```math -F(x;\nu) = 1 - \frac{1}{2} \frac{\operatorname{Beta}(\tfrac{\nu}{\nu + x^2};\,\tfrac{\nu}{2},\tfrac{1}{2})}{\operatorname{Beta}(\tfrac{\nu}{2}, \tfrac{1}{2})} +F(x;\nu) = 1 - \frac{1}{2} \frac{\mathop{\mathrm{Beta}}(\tfrac{\nu}{\nu + x^2};\,\tfrac{\nu}{2},\tfrac{1}{2})}{\mathop{\mathrm{Beta}}(\tfrac{\nu}{2}, \tfrac{1}{2})} ``` ```math -\operatorname{Kurt}\left( X \right) = \begin{cases} \frac{6}{\nu-4} & \text{ for } \nu > 2 \\ \infty & \text{ for } 2 < \nu \le 4 \end{cases} +\mathop{\mathrm{Kurt}}\left( X \right) = \begin{cases} \frac{6}{\nu-4} & \text{ for } \nu > 2 \\ \infty & \text{ for } 2 < \nu \le 4 \end{cases} ``` ```math -F(x;\nu) = 1 - \frac{1}{2} \frac{\operatorname{Beta}(\tfrac{\nu}{\nu + x^2};\,\tfrac{\nu}{2},\tfrac{1}{2})}{\operatorname{Beta}(\tfrac{\nu}{2}, \tfrac{1}{2})} +F(x;\nu) = 1 - \frac{1}{2} \frac{\mathop{\mathrm{Beta}}(\tfrac{\nu}{\nu + x^2};\,\tfrac{\nu}{2},\tfrac{1}{2})}{\mathop{\mathrm{Beta}}(\tfrac{\nu}{2}, \tfrac{1}{2})} ``` ```math -\operatorname{Median}\left( X \right) = 0 +\mathop{\mathrm{Median}}\left( X \right) = 0 ``` ```math -\operatorname{mode}\left( X \right) = 0 +\mathop{\mathrm{mode}}\left( X \right) = 0 ``` ```math -\operatorname{skew}\left( X \right) = 0 +\mathop{\mathrm{skew}}\left( X \right) = 0 ``` ```math -\operatorname{Var}\left( X \right) = \begin{cases} \frac{\nu }{\nu-2} & \text{ for } \nu > 2 \\ \infty & \text{ for } 1 < \nu \le 2 \end{cases} +\mathop{\mathrm{Var}}\left( X \right) = \begin{cases} \frac{\nu }{\nu-2} & \text{ for } \nu > 2 \\ \infty & \text{ for } 1 < \nu \le 2 \end{cases} ``` ```math -\operatorname{Kurt}\left( X \right) = -\frac{3}{5} +\mathop{\mathrm{Kurt}}\left( X \right) = -\frac{3}{5} ``` ```math -\operatorname{Median}\left( X \right) = \begin{cases}a+{\sqrt {\frac {(b-a)(c-a)}{2}}}&{\text{ for }}c\geq {\frac {a+b}{2}}\\[6pt]b-{\sqrt {\frac {(b-a)(b-c)}{2}}}&{\text{ for }}c\leq {\frac{a+b}{2}}\end{cases} +\mathop{\mathrm{Median}}\left( X \right) = \begin{cases}a+{\sqrt {\frac {(b-a)(c-a)}{2}}}&{\text{ for }}c\geq {\frac {a+b}{2}}\\[6pt]b-{\sqrt {\frac {(b-a)(b-c)}{2}}}&{\text{ for }}c\leq {\frac{a+b}{2}}\end{cases} ``` ```math -\operatorname{mode}\left( X \right) = c +\mathop{\mathrm{mode}}\left( X \right) = c ``` ```math -\operatorname{skew}\left( X \right) = \frac{{\sqrt 2}(a\!+\!b\!-\!2c)(2a\!-\!b\!-\!c)(a\!-\!2b\!+\!c)}{5(a^{2}\!+\!b^{2}\!+\!c^{2}\!-\!ab\!-\!ac\!-\!bc)^{{\frac{3}{2}}}} +\mathop{\mathrm{skew}}\left( X \right) = \frac{{\sqrt 2}(a\!+\!b\!-\!2c)(2a\!-\!b\!-\!c)(a\!-\!2b\!+\!c)}{5(a^{2}\!+\!b^{2}\!+\!c^{2}\!-\!ab\!-\!ac\!-\!bc)^{{\frac{3}{2}}}} ``` ```math -\operatorname{Var}\left( X \right) = \frac{a^{2}+b^{2}+c^{2}-ab-ac-bc}{18} +\mathop{\mathrm{Var}}\left( X \right) = \frac{a^{2}+b^{2}+c^{2}-ab-ac-bc}{18} ``` ```math -\operatorname{Kurt}\left( X \right) = -{\tfrac{6}{5}} +\mathop{\mathrm{Kurt}}\left( X \right) = -{\tfrac{6}{5}} ``` ```math -\operatorname{Median}\left[ X \right] = \frac{1}{2} \left( a + b \right) +\mathop{\mathrm{Median}}\left[ X \right] = \frac{1}{2} \left( a + b \right) ``` ```math -\operatorname{skew}\left[ X \right] = 0 +\mathop{\mathrm{skew}}\left[ X \right] = 0 ``` ```math -\operatorname{Var}\left( X \right) = \tfrac{1}{12} \left( b - a \right)^2 +\mathop{\mathrm{Var}}\left( X \right) = \tfrac{1}{12} \left( b - a \right)^2 ``` ```math -\operatorname{Kurt}\left( X \right) = \frac{-6\Gamma_1^4+12\Gamma_1^2\Gamma_2-3\Gamma_2^2 -4\Gamma_1\Gamma_3+\Gamma_4}{[\Gamma_2-\Gamma_1^2]^2} +\mathop{\mathrm{Kurt}}\left( X \right) = \frac{-6\Gamma_1^4+12\Gamma_1^2\Gamma_2-3\Gamma_2^2 -4\Gamma_1\Gamma_3+\Gamma_4}{[\Gamma_2-\Gamma_1^2]^2} ``` ```math -\operatorname{Median}\left( X \right) = \lambda(\ln(2))^{1/k} +\mathop{\mathrm{Median}}\left( X \right) = \lambda(\ln(2))^{1/k} ``` ```math -\operatorname{mode}\left( X \right) = {\displaystyle {\begin{cases}\lambda \left({\frac {k-1}{k}}\right)^{\frac {1}{k}}\,&k>1\\0&k\leq 1\end{cases}}} +\mathop{\mathrm{mode}}\left( X \right) = {\displaystyle {\begin{cases}\lambda \left({\frac {k-1}{k}}\right)^{\frac {1}{k}}\,&k>1\\0&k\leq 1\end{cases}}} ``` ```math -\operatorname{skew}\left( X \right) = \frac{\Gamma(1+3/k)\lambda^3-3\mu\sigma^2-\mu^3}{\sigma^3} +\mathop{\mathrm{skew}}\left( X \right) = \frac{\Gamma(1+3/k)\lambda^3-3\mu\sigma^2-\mu^3}{\sigma^3} ``` ```math -\operatorname{Var}\left( X \right) = \lambda^2\left[\Gamma\left(1+\frac{2}{k}\right) - \left(\Gamma\left(1+\frac{1}{k}\right)\right)^2\right] +\mathop{\mathrm{Var}}\left( X \right) = \lambda^2\left[\Gamma\left(1+\frac{2}{k}\right) - \left(\Gamma\left(1+\frac{1}{k}\right)\right)^2\right] ``` ```math -\rho_{X,Y} = \frac{\operatorname{cov}(X,Y)}{\sigma_X \sigma_Y} +\rho_{X,Y} = \frac{\mathop{\mathrm{cov}}(X,Y)}{\sigma_X \sigma_Y} ``` ```math -\operatorname{cov_n} = \frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x}_n)(y_i - \bar{y}_n) +\mathop{\mathrm{cov_n}} = \frac{1}{n-1} \sum_{i=0}^{n-1} (x_i - \bar{x}_n)(y_i - \bar{y}_n) ``` ```math -\operatorname{cov_n} = \frac{1}{n} \sum_{i=0}^{n-1} (x_i - \mu_x)(y_i - \mu_y) +\mathop{\mathrm{cov_n}} = \frac{1}{n} \sum_{i=0}^{n-1} (x_i - \mu_x)(y_i - \mu_y) ``` ```math -\operatorname{cov_{jkn}} = \frac{1}{n-1} \sum_{i=0}^{n-1} (x_{ij} - \bar{x}_{jn})(x_{ik} - \bar{x}_{kn}) +\mathop{\mathrm{cov_{jkn}}} = \frac{1}{n-1} \sum_{i=0}^{n-1} (x_{ij} - \bar{x}_{jn})(x_{ik} - \bar{x}_{kn}) ``` ```math -\operatorname{cov_{jkn}} = \frac{1}{n} \sum_{i=0}^{n-1} (x_{ij} - \mu_{j})(x_{ik} - \mu_{k}) +\mathop{\mathrm{cov_{jkn}}} = \frac{1}{n} \sum_{i=0}^{n-1} (x_{ij} - \mu_{j})(x_{ik} - \mu_{k}) ``` ```math -\operatorname{Kurtosis}[X] = \mathrm{E}\biggl[ \biggl( \frac{X - \mu}{\sigma} \biggr)^4 \biggr] +\mathop{\mathrm{Kurtosis}}[X] = \mathrm{E}\biggl[ \biggl( \frac{X - \mu}{\sigma} \biggr)^4 \biggr] ``` ```math -\operatorname{MAAPE} = \frac{1}{n} \sum_{i=0}^{n-1} \operatorname{arctan}\biggl( \biggl| \frac{a_i - f_i}{a_i} \biggr| \biggr) +\mathop{\mathrm{MAAPE}} = \frac{1}{n} \sum_{i=0}^{n-1} \mathop{\mathrm{arctan}}\biggl( \biggl| \frac{a_i - f_i}{a_i} \biggr| \biggr) ``` ```math -\operatorname{MAE} = \frac{\displaystyle\sum_{i=0}^{n-1} |y_i - x_i|}{n} +\mathop{\mathrm{MAE}} = \frac{\displaystyle\sum_{i=0}^{n-1} |y_i - x_i|}{n} ``` ```math -\rho_{X,Y} = \frac{\operatorname{cov}(X,Y)}{\sigma_X \sigma_Y} +\rho_{X,Y} = \frac{\mathop{\mathrm{cov}}(X,Y)}{\sigma_X \sigma_Y} ``` ```math -\operatorname{MAPE} = \frac{100}{n} \sum_{i=0}^{n-1} \biggl| \frac{a_i - f_i}{a_i} \biggr| +\mathop{\mathrm{MAPE}} = \frac{100}{n} \sum_{i=0}^{n-1} \biggl| \frac{a_i - f_i}{a_i} \biggr| ``` ```math -\operatorname{cov_n} = \frac{1}{n-1} \sum_{i=j}^{j+W-1} (x_i - \bar{x}_n)(y_i - \bar{y}_n) +\mathop{\mathrm{cov_n}} = \frac{1}{n-1} \sum_{i=j}^{j+W-1} (x_i - \bar{x}_n)(y_i - \bar{y}_n) ``` ```math -\operatorname{cov_n} = \frac{1}{n} \sum_{i=j}^{j+W-1} (x_i - \mu_x)(y_i - \mu_y) +\mathop{\mathrm{cov_n}} = \frac{1}{n} \sum_{i=j}^{j+W-1} (x_i - \mu_x)(y_i - \mu_y) ``` ```math -\operatorname{MDA} = \begin{cases} 1 & \textrm{if}\ N = 1 \\\frac{1}{N} \sum_{i=1}^{N} \delta_{\mathop{\mathrm{sgn}}(\Delta f_{i,i-1}),\ \mathop{\mathrm{sgn}}(\Delta a_{i,i-1})} & \textrm{if}\ N > 1 \end{cases} +\mathop{\mathrm{MDA}} = \begin{cases} 1 & \textrm{if}\ N = 1 \\\frac{1}{N} \sum_{i=1}^{N} \delta_{\mathop{\mathrm{sgn}}(\Delta f_{i,i-1}),\ \mathop{\mathrm{sgn}}(\Delta a_{i,i-1})} & \textrm{if}\ N > 1 \end{cases} ``` ```math -\operatorname{ME} = \frac{1}{n} \sum_{i=0}^{n-1} (y_i - x_i) +\mathop{\mathrm{ME}} = \frac{1}{n} \sum_{i=0}^{n-1} (y_i - x_i) ``` ```math -\operatorname{MAAPE} = \frac{1}{W} \sum_{i=0}^{W-1} \operatorname{arctan}\biggl( \biggl| \frac{a_i - f_i}{a_i} \biggr| \biggr) +\mathop{\mathrm{MAAPE}} = \frac{1}{W} \sum_{i=0}^{W-1} \mathop{\mathrm{arctan}}\biggl( \biggl| \frac{a_i - f_i}{a_i} \biggr| \biggr) ``` ```math -\operatorname{MAE} = \frac{1}{W} \sum_{i=0}^{W-1} |y_i - x_i| +\mathop{\mathrm{MAE}} = \frac{1}{W} \sum_{i=0}^{W-1} |y_i - x_i| ``` ```math -\operatorname{MAPE} = \frac{100}{W} \sum_{i=0}^{W-1} \biggl| \frac{a_i - f_i}{a_i} \biggr| +\mathop{\mathrm{MAPE}} = \frac{100}{W} \sum_{i=0}^{W-1} \biggl| \frac{a_i - f_i}{a_i} \biggr| ``` ```math -\operatorname{MDA} = \begin{cases} 1 & \textrm{if}\ W = 1 \\ \frac{1}{W} \sum_{i=1}^{W} \delta_{\operatorname{sgn}(\Delta f_{i,i-1}),\ \operatorname{sgn}(\Delta a_{i,i-1})} & \textrm{if}\ W > 1 \end{cases} +\mathop{\mathrm{MDA}} = \begin{cases} 1 & \textrm{if}\ W = 1 \\ \frac{1}{W} \sum_{i=1}^{W} \delta_{\mathop{\mathrm{sgn}}(\Delta f_{i,i-1}),\ \mathop{\mathrm{sgn}}(\Delta a_{i,i-1})} & \textrm{if}\ W > 1 \end{cases} ``` ```math -\operatorname{ME} = \frac{1}{W} \sum_{i=0}^{W-1} (y_i - x_i) +\mathop{\mathrm{ME}} = \frac{1}{W} \sum_{i=0}^{W-1} (y_i - x_i) ``` ```math -\operatorname{MPE} = \frac{100}{W} \sum_{i=0}^{W-1} \frac{a_i - f_i}{a_i} +\mathop{\mathrm{MPE}} = \frac{100}{W} \sum_{i=0}^{W-1} \frac{a_i - f_i}{a_i} ``` ```math -\operatorname{MSE} = \frac{1}{W} \sum_{i=0}^{W-1} (y_i - x_i)^2 +\mathop{\mathrm{MSE}} = \frac{1}{W} \sum_{i=0}^{W-1} (y_i - x_i)^2 ``` ```math -\rho_{X,Y} = \frac{\operatorname{cov}(X,Y)}{\sigma_X \sigma_Y} +\rho_{X,Y} = \frac{\mathop{\mathrm{cov}}(X,Y)}{\sigma_X \sigma_Y} ``` ```math -\rho_{X,Y} = \frac{\operatorname{cov}(X,Y)}{\sigma_X \sigma_Y} +\rho_{X,Y} = \frac{\mathop{\mathrm{cov}}(X,Y)}{\sigma_X \sigma_Y} ``` ```math -d_{x,y} = 1 - r_{x,y} = 1 - \frac{\operatorname{cov_n(x,y)}}{\sigma_x \sigma_y} +d_{x,y} = 1 - r_{x,y} = 1 - \frac{\mathop{\mathrm{cov_n(x,y)}}}{\sigma_x \sigma_y} ``` ```math -\operatorname{MPE} = \frac{100}{n} \sum_{i=0}^{n-1} \frac{a_i - f_i}{a_i} +\mathop{\mathrm{MPE}} = \frac{100}{n} \sum_{i=0}^{n-1} \frac{a_i - f_i}{a_i} ``` ```math -\operatorname{RMSE} = \sqrt{ \frac{1}{W} \sum_{i=0}^{W-1} (y_i - x_i)^2 } +\mathop{\mathrm{RMSE}} = \sqrt{ \frac{1}{W} \sum_{i=0}^{W-1} (y_i - x_i)^2 } ``` ```math -\operatorname{RSS} = \sum_{i=0}^{W-1} (y_i - x_i)^2 +\mathop{\mathrm{RSS}} = \sum_{i=0}^{W-1} (y_i - x_i)^2 ``` ```math -\operatorname{MSE} = \frac{1}{n} \sum_{i=0}^{n-1} (y_i - x_i)^2 +\mathop{\mathrm{MSE}} = \frac{1}{n} \sum_{i=0}^{n-1} (y_i - x_i)^2 ``` ```math -\operatorname{RMSE} = \sqrt{ \frac{1}{n} \sum_{i=0}^{n-1} (y_i - x_i)^2 } +\mathop{\mathrm{RMSE}} = \sqrt{ \frac{1}{n} \sum_{i=0}^{n-1} (y_i - x_i)^2 } ``` ```math -\operatorname{RSS} = \sum_{i=0}^{n-1} (y_i - x_i)^2 +\mathop{\mathrm{RSS}} = \sum_{i=0}^{n-1} (y_i - x_i)^2 ``` ```math -\operatorname{Skewness}[X] = \mathrm{E}\biggl[ \biggl( \frac{X - \mu}{\sigma} \biggr)^3 \biggr] +\mathop{\mathrm{Skewness}}[X] = \mathrm{E}\biggl[ \biggl( \frac{X - \mu}{\sigma} \biggr)^3 \biggr] ```