Mean and variance of beta looking distribution
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From lecture notes on SDE's. I have calculated a stationary distribution of an SDE from the forward kolmogorov equation. I can not, however, identify this distribution and I want to find the mean and variance. The pdf is.
$f(x)=frac{(x^2+1)^{-left(frac{sigma^2+lambda}{sigma^2} right)}Gammaleft( frac{sigma^2+lambda}{sigma^2}right)}{sqrt{pi}Gammaleft(-frac{1}{2}+frac{sigma^2+lambda}{sigma^2} right)}$
Where $lambda,sigma$ are real scalars and $Gamma$ is the gamma function. It looks like a beta-distribution, but I cannot wrangle it into a form where it yields it's mean and variance?
probability probability-theory probability-distributions stochastic-processes stochastic-calculus
asked Dec 18 '18 at 23:54
thaumoctopusthaumoctopus
9519
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add a comment |
$begingroup$
From lecture notes on SDE's. I have calculated a stationary distribution of an SDE from the forward kolmogorov equation. I can not, however, identify this distribution and I want to find the mean and variance. The pdf is.
$f(x)=frac{(x^2+1)^{-left(frac{sigma^2+lambda}{sigma^2} right)}Gammaleft( frac{sigma^2+lambda}{sigma^2}right)}{sqrt{pi}Gammaleft(-frac{1}{2}+frac{sigma^2+lambda}{sigma^2} right)}$
Where $lambda,sigma$ are real scalars and $Gamma$ is the gamma function. It looks like a beta-distribution, but I cannot wrangle it into a form where it yields it's mean and variance?
probability probability-theory probability-distributions stochastic-processes stochastic-calculus
asked Dec 18 '18 at 23:54
thaumoctopusthaumoctopus
9519
$endgroup$
1
$begingroup$
It's a Student $t$, actually. That'll give what you need.
$endgroup$
– J.G.
Dec 19 '18 at 0:19
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That is exactly what it is, thank you so much!
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– thaumoctopus
Dec 19 '18 at 0:20
1
$begingroup$
I think strictly it may be a scaled Student $t$. In any case if $lambda=0$ it is a Cauchy distribution with no mean, and if $0 lt lambda le sigma^2$ then it has no variance
$endgroup$
– Henry
Dec 19 '18 at 0:22
add a comment |
$begingroup$
From lecture notes on SDE's. I have calculated a stationary distribution of an SDE from the forward kolmogorov equation. I can not, however, identify this distribution and I want to find the mean and variance. The pdf is.
$f(x)=frac{(x^2+1)^{-left(frac{sigma^2+lambda}{sigma^2} right)}Gammaleft( frac{sigma^2+lambda}{sigma^2}right)}{sqrt{pi}Gammaleft(-frac{1}{2}+frac{sigma^2+lambda}{sigma^2} right)}$
Where $lambda,sigma$ are real scalars and $Gamma$ is the gamma function. It looks like a beta-distribution, but I cannot wrangle it into a form where it yields it's mean and variance?
probability probability-theory probability-distributions stochastic-processes stochastic-calculus
asked Dec 18 '18 at 23:54
thaumoctopusthaumoctopus
9519
$endgroup$
From lecture notes on SDE's. I have calculated a stationary distribution of an SDE from the forward kolmogorov equation. I can not, however, identify this distribution and I want to find the mean and variance. The pdf is.
$f(x)=frac{(x^2+1)^{-left(frac{sigma^2+lambda}{sigma^2} right)}Gammaleft( frac{sigma^2+lambda}{sigma^2}right)}{sqrt{pi}Gammaleft(-frac{1}{2}+frac{sigma^2+lambda}{sigma^2} right)}$
Where $lambda,sigma$ are real scalars and $Gamma$ is the gamma function. It looks like a beta-distribution, but I cannot wrangle it into a form where it yields it's mean and variance?
probability probability-theory probability-distributions stochastic-processes stochastic-calculus
probability probability-theory probability-distributions stochastic-processes stochastic-calculus
asked Dec 18 '18 at 23:54
thaumoctopusthaumoctopus
9519
asked Dec 18 '18 at 23:54
thaumoctopusthaumoctopus
9519
edited Dec 19 '18 at 0:11
thaumoctopus
asked Dec 18 '18 at 23:54
thaumoctopusthaumoctopus
9519
asked Dec 18 '18 at 23:54
thaumoctopusthaumoctopus
9519
asked Dec 18 '18 at 23:54
thaumoctopusthaumoctopus
9519
9519
1
$begingroup$
It's a Student $t$, actually. That'll give what you need.
$endgroup$
– J.G.
Dec 19 '18 at 0:19
$begingroup$
That is exactly what it is, thank you so much!
$endgroup$
– thaumoctopus
Dec 19 '18 at 0:20
1
$begingroup$
I think strictly it may be a scaled Student $t$. In any case if $lambda=0$ it is a Cauchy distribution with no mean, and if $0 lt lambda le sigma^2$ then it has no variance
$endgroup$
– Henry
Dec 19 '18 at 0:22
add a comment |
1
$begingroup$
It's a Student $t$, actually. That'll give what you need.
$endgroup$
– J.G.
Dec 19 '18 at 0:19
$begingroup$
That is exactly what it is, thank you so much!
$endgroup$
– thaumoctopus
Dec 19 '18 at 0:20
1
$begingroup$
I think strictly it may be a scaled Student $t$. In any case if $lambda=0$ it is a Cauchy distribution with no mean, and if $0 lt lambda le sigma^2$ then it has no variance
$endgroup$
– Henry
Dec 19 '18 at 0:22
1
1
$begingroup$
It's a Student $t$, actually. That'll give what you need.
$endgroup$
– J.G.
Dec 19 '18 at 0:19
$begingroup$
It's a Student $t$, actually. That'll give what you need.
$endgroup$
– J.G.
Dec 19 '18 at 0:19
$begingroup$
That is exactly what it is, thank you so much!
$endgroup$
– thaumoctopus
Dec 19 '18 at 0:20
$begingroup$
That is exactly what it is, thank you so much!
$endgroup$
– thaumoctopus
Dec 19 '18 at 0:20
1
1
$begingroup$
I think strictly it may be a scaled Student $t$. In any case if $lambda=0$ it is a Cauchy distribution with no mean, and if $0 lt lambda le sigma^2$ then it has no variance
$endgroup$
– Henry
Dec 19 '18 at 0:22
$begingroup$
I think strictly it may be a scaled Student $t$. In any case if $lambda=0$ it is a Cauchy distribution with no mean, and if $0 lt lambda le sigma^2$ then it has no variance
$endgroup$
– Henry
Dec 19 '18 at 0:22
add a comment |
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1
$begingroup$
It's a Student $t$, actually. That'll give what you need.
$endgroup$
– J.G.
Dec 19 '18 at 0:19
$begingroup$
That is exactly what it is, thank you so much!
$endgroup$
– thaumoctopus
Dec 19 '18 at 0:20
1
$begingroup$
I think strictly it may be a scaled Student $t$. In any case if $lambda=0$ it is a Cauchy distribution with no mean, and if $0 lt lambda le sigma^2$ then it has no variance
$endgroup$
– Henry
Dec 19 '18 at 0:22