Revision #2569 → #3167 · back to history
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| note | `expMeasure r` is the exponential measure on ℝ with rate `r`, defined via `gammaMeasure 1 r`. | `expMeasure r` is the exponential measure on ℝ with rate `r`, defined as `gammaMeasure 1 r`. |
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| anchors | [{"section":"Alternative parametrization","snippet":"The exponential distribution is sometimes parametrized in terms of the scale parameter"},{"type":"math_alttext","value":"{\\displaystyle f(x;\\beta )={\\begin{cases}{\\frac {1}{\\beta }}e^{-x/\\beta }&x\\geq 0,\\\\0&x<0.\\end{cases}}\\qquad \\qquad F(x;\\beta )={\\begin{cases}1-e^{-x/\\beta }&x\\geq 0,\\\\0&x<0.\\end{cases}}}"}] | [{"section":"Alternative parametrization","snippet":"The exponential distribution is sometimes parametrized in terms of the scale parameter"},{"type":"math_alttext","value":"{\\displaystyle f(x;\\beta )={\\begin{cases}{\\frac {1}{\\beta }}e^{-x/\\beta }&x\\geq 0,\\\\0&x<0.\\end{cases}}\\qquad \\qquad F(x;\\beta )={\\begin{cases}1-e^{-x/\\beta }&x\\geq 0,\\\\0&x<0.\\end{cases}}}"}] |
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| anchors | [{"section":"Mean, variance, moments, and median","snippet":"The median of X is given by"},{"type":"math_alttext","value":"{\\displaystyle \\operatorname {m} [X]={\\frac {\\ln(2)}{\\lambda }}<\\operatorname {E} [X],}"},{"type":"math_alttext","value":"{\\displaystyle \\left|\\operatorname {E} \\left[X\\right]-\\operatorname {m} \\left[X\\right]\\right|={\\frac {1-\\ln(2)}{\\lambda }}<{\\frac {1}{\\lambda }}=\\operatorname {\\sigma } [X],}"}] | [{"section":"Mean, variance, moments, and median","snippet":"The median of X is given by"},{"type":"math_alttext","value":"{\\displaystyle \\operatorname {m} [X]={\\frac {\\ln(2)}{\\lambda }}<\\operatorname {E} [X],}"},{"type":"math_alttext","value":"{\\displaystyle \\left|\\operatorname {E} \\left[X\\right]-\\operatorname {m} \\left[X\\right]\\right|={\\frac {1-\\ln(2)}{\\lambda }}<{\\frac {1}{\\lambda }}=\\operatorname {\\sigma } [X],}"}] |
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| anchors | [{"section":"Memorylessness property of exponential random variable","snippet":"An exponentially distributed random variable T obeys the relation"},{"type":"math_alttext","value":"{\\displaystyle \\Pr \\left(T>s+t\\mid T>s\\right)=\\Pr(T>t),\\qquad \\forall s,t\\geq 0.}"}] | [{"section":"Memorylessness property of exponential random variable","snippet":"An exponentially distributed random variable T obeys the relation"},{"type":"math_alttext","value":"{\\displaystyle \\Pr \\left(T>s+t\\mid T>s\\right)=\\Pr(T>t),\\qquad \\forall s,t\\geq 0.}"}] |
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| anchors | [{"section":"Quantiles","snippet":"The quantile function (inverse cumulative distribution function) for Exp"},{"type":"math_alttext","value":"{\\displaystyle F^{-1}(p;\\lambda )={\\frac {-\\ln(1-p)}{\\lambda }},\\qquad 0\\leq p<1}"}] | [{"section":"Quantiles","snippet":"The quantile function (inverse cumulative distribution function) for Exp"},{"type":"math_alttext","value":"{\\displaystyle F^{-1}(p;\\lambda )={\\frac {-\\ln(1-p)}{\\lambda }},\\qquad 0\\leq p<1}"}] |
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| anchors | [{"section":"Sum of two independent exponential random variables","snippet":"are independent exponential random variables with respective rate parameters"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}f_{Z}(z)&=\\int _{-\\infty }^{\\infty }f_{X_{1}}(x_{1})f_{X_{2}}(z-x_{1})\\,dx_{1}\\\\&=\\int _{0}^{z}\\lambda _{1}e^{-\\lambda _{1}x_{1}}\\lambda _{2}e^{-\\lambda _{2}(z-x_{1})}\\,dx_{1}\\\\&=\\lambda _{1}\\lambda _{2}e^{-\\lambda _{2}z}\\int _{0}^{z}e^{(\\lambda _{2}-\\lambda _{1})x_{1}}\\,dx_{1}\\\\&={\\begin{cases}{\\dfrac {\\lambda _{1}\\lambda _{2}}{\\lambda _{2}-\\lambda _{1}}}\\left(e^{-\\lambda _{1}z}-e^{-\\lambda _{2}z}\\right)&{\\text{ if }}\\lambda _{1}\\neq \\lambda _{2}\\\\[4pt]\\lambda ^{2}ze^{-\\lambda z}&{\\text{ if }}\\lambda _{1}=\\lambda _{2}=\\lambda .\\end{cases}}\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}H(Z)&=1+\\gamma +\\ln \\left({\\frac {\\lambda _{1}-\\lambda _{2}}{\\lambda _{1}\\lambda _{2}}}\\right)+\\psi \\left({\\frac {\\lambda _{1}}{\\lambda _{1}-\\lambda _{2}}}\\right),\\end{aligned}}}"}] | [{"section":"Sum of two independent exponential random variables","snippet":"are independent exponential random variables with respective rate parameters"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}f_{Z}(z)&=\\int _{-\\infty }^{\\infty }f_{X_{1}}(x_{1})f_{X_{2}}(z-x_{1})\\,dx_{1}\\\\&=\\int _{0}^{z}\\lambda _{1}e^{-\\lambda _{1}x_{1}}\\lambda _{2}e^{-\\lambda _{2}(z-x_{1})}\\,dx_{1}\\\\&=\\lambda _{1}\\lambda _{2}e^{-\\lambda _{2}z}\\int _{0}^{z}e^{(\\lambda _{2}-\\lambda _{1})x_{1}}\\,dx_{1}\\\\&={\\begin{cases}{\\dfrac {\\lambda _{1}\\lambda _{2}}{\\lambda _{2}-\\lambda _{1}}}\\left(e^{-\\lambda _{1}z}-e^{-\\lambda _{2}z}\\right)&{\\text{ if }}\\lambda _{1}\\neq \\lambda _{2}\\\\[4pt]\\lambda ^{2}ze^{-\\lambda z}&{\\text{ if }}\\lambda _{1}=\\lambda _{2}=\\lambda .\\end{cases}}\\end{aligned}}}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}H(Z)&=1+\\gamma +\\ln \\left({\\frac {\\lambda _{1}-\\lambda _{2}}{\\lambda _{1}\\lambda _{2}}}\\right)+\\psi \\left({\\frac {\\lambda _{1}}{\\lambda _{1}-\\lambda _{2}}}\\right),\\end{aligned}}"}] |
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| anchors | [{"section":"Confidence intervals","snippet":"An exact 100(1 − α)% confidence interval for the rate parameter of an exponential distribution is given by"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {2n}{{\\widehat {\\lambda }}_{\\textrm {mle}}\\chi _{{\\frac {\\alpha }{2}},2n}^{2}}}<{\\frac {1}{\\lambda }}<{\\frac {2n}{{\\widehat {\\lambda }}_{\\textrm {mle}}\\chi _{1-{\\frac {\\alpha }{2}},2n}^{2}}}\\,,}"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {2n{\\overline {x}}}{\\chi _{{\\frac {\\alpha }{2}},2n}^{2}}}<{\\frac {1}{\\lambda }}<{\\frac {2n{\\overline {x}}}{\\chi _{1-{\\frac {\\alpha }{2}},2n}^{2}}}\\,,}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\lambda _{\\text{lower}}&={\\widehat {\\lambda }}\\left(1-{\\frac {1.96}{\\sqrt {n}}}\\right)\\\\\\lambda _{\\text{upper}}&={\\widehat {\\lambda }}\\left(1+{\\frac {1.96}{\\sqrt {n}}}\\right)\\end{aligned}}}"}] | [{"section":"Confidence intervals","snippet":"An exact 100(1 − α)% confidence interval for the rate parameter of an exponential distribution is given by"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {2n}{{\\widehat {\\lambda }}_{\\textrm {mle}}\\chi _{{\\frac {\\alpha }{2}},2n}^{2}}}<{\\frac {1}{\\lambda }}<{\\frac {2n}{{\\widehat {\\lambda }}_{\\textrm {mle}}\\chi _{1-{\\frac {\\alpha }{2}},2n}^{2}}}\\,,}"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {2n{\\overline {x}}}{\\chi _{{\\frac {\\alpha }{2}},2n}^{2}}}<{\\frac {1}{\\lambda }}<{\\frac {2n{\\overline {x}}}{\\chi _{1-{\\frac {\\alpha }{2}},2n}^{2}}}\\,,}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\lambda _{\\text{lower}}&={\\widehat {\\lambda }}\\left(1-{\\frac {1.96}{\\sqrt {n}}}\\right)\\\\\\lambda _{\\text{upper}}&={\\widehat {\\lambda }}\\left(1+{\\frac {1.96}{\\sqrt {n}}}\\right)\\end{aligned}}"}] |
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| anchors | [{"section":"Confidence intervals","snippet":"A simple approximation to the exact interval endpoints can be derived using a normal approximation"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {2n}{{\\widehat {\\lambda }}_{\\textrm {mle}}\\chi _{{\\frac {\\alpha }{2}},2n}^{2}}}<{\\frac {1}{\\lambda }}<{\\frac {2n}{{\\widehat {\\lambda }}_{\\textrm {mle}}\\chi _{1-{\\frac {\\alpha }{2}},2n}^{2}}}\\,,}"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {2n{\\overline {x}}}{\\chi _{{\\frac {\\alpha }{2}},2n}^{2}}}<{\\frac {1}{\\lambda }}<{\\frac {2n{\\overline {x}}}{\\chi _{1-{\\frac {\\alpha }{2}},2n}^{2}}}\\,,}"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\lambda _{\\text{lower}}&={\\widehat {\\lambda }}\\left(1-{\\frac {1.96}{\\sqrt {n}}}\\right)\\\\\\lambda _{\\text{upper}}&={\\widehat {\\lambda }}\\left(1+{\\frac {1.96}{\\sqrt {n}}}\\right)\\end{aligned}}}"}] | [{"section":"Confidence intervals","snippet":"A simple approximation to the exact interval endpoints can be derived using a normal approximation"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {2n}{{\\widehat {\\lambda }}_{\\textrm {mle}}\\chi _{{\\frac {\\alpha }{2}},2n}^{2}}}<{\\frac {1}{\\lambda }}<{\\frac {2n}{{\\widehat {\\lambda }}_{\\textrm {mle}}\\chi _{1-{\\frac {\\alpha }{2}},2n}^{2}}}\\,,}"},{"type":"math_alttext","value":"{\\displaystyle {\\frac {2n{\\overline {x}}}{\\chi _{{\\frac {\\alpha }{2}},2n}^{2}}}<{\\frac {1}{\\lambda }}<{\\frac {2n{\\overline {x}}}{\\chi _{1-{\\frac {\\alpha }{2}},2n}^{2}}}\\,,"},{"type":"math_alttext","value":"{\\displaystyle {\\begin{aligned}\\lambda _{\\text{lower}}&={\\widehat {\\lambda }}\\left(1-{\\frac {1.96}{\\sqrt {n}}}\\right)\\\\\\lambda _{\\text{upper}}&={\\widehat {\\lambda }}\\left(1+{\\frac {1.96}{\\sqrt {n}}}\\right)\\end{aligned}}}"}] |
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