47 lines
1.9 KiB
ReStructuredText
47 lines
1.9 KiB
ReStructuredText
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.. _continuous-triang:
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Triangular Distribution
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=======================
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One shape parameter :math:`c\in[0,1]` giving the distance to the peak as a percentage of the total extent of
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the non-zero portion. The location parameter is the start of the non-
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zero portion, and the scale-parameter is the width of the non-zero
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portion. In standard form we have :math:`x\in\left[0,1\right].`
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.. math::
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:nowrap:
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\begin{eqnarray*}
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f\left(x;c\right) & = & \left\{
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\begin{array}{ccc}
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2\frac{x}{c} & & x < c \\
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2\frac{1-x}{1-c} & & x \geq c
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\end{array}
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\right.\\
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F\left(x;c\right) & = & \left\{
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\begin{array}{ccc}
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\frac{x^{2}}{c} & & x < c \\
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\frac{x^{2}-2x+c}{c-1} & & x \geq c
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\end{array}
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\right.\\
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G\left(q;c\right) & = & \left\{
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\begin{array}{ccc}
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\sqrt{cq} & & q < c \\
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1-\sqrt{\left(1-c\right)\left(1-q\right)} & & q \geq c
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\end{array}
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\right.
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\end{eqnarray*}
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.. math::
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:nowrap:
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\begin{eqnarray*} \mu & = & \frac{c}{3}+\frac{1}{3}\\ \mu_{2} & = & \frac{1-c+c^{2}}{18}\\ \gamma_{1} & = & \frac{\sqrt{2}\left(2c-1\right)\left(c+1\right)\left(c-2\right)}{5\left(1-c+c^{2}\right)^{3/2}}\\ \gamma_{2} & = & -\frac{3}{5}\end{eqnarray*}
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.. math::
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:nowrap:
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\begin{eqnarray*} h\left(X\right) & = & \log\left(\frac{1}{2}\sqrt{e}\right)\\ & \approx & -0.19314718055994530942.\end{eqnarray*}
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Implementation: `scipy.stats.triang`
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