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Compute Variance and Skewness for my below PDF ?

Posted 11 months ago
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I have tried many times to compute Variance and mean of the following PDF using Wolfram cloud But I failed ;$$ F(z,\mu,\sigma)=\frac{2 (z-\sigma )^2 \exp \left(-\frac{(z-\sigma )^2 \sqrt{\left(1+0.25 \mu ^2\right) 2 \pi } \text{erf}\left(\frac{(z-\sigma )^2 \sqrt{\left(1+0.25 \mu ^2\right) 2 \pi }}{1+0.25 \mu ^2}\right)}{1+0.25 \mu ^2}\right)}{\pi ^2 \sqrt{\left(1+0.25 \mu ^2\right) 2 \pi }} \,$$

Note: $\mu \in (0,1)$, $z , \sigma$ are reals .

This is My CODE :

integrand[z_, \[Mu]_, \[Sigma]_]:=(2/Pi^2)*(z-\[Sigma])^2/(Sqrt[(1+0.25\[Mu]^2)2*Pi])Exp[-(z-\[Sigma])^2/(1+0.25\[Mu]^2)*Sqrt[(1+0.25\[Mu]^2)*2*Pi]
* Erf[(z-\[Sigma])^2/(1+0.25\[Mu]^2)Sqrt[(1+0.25\[Mu]^2)*2*Pi] ]]
\[ScriptCapitalD]= ProbabilityDistribution[f[x,\[Mu] ], {x,0, \[Infinity]}]
pdfF[\[Mu]_?NumericQ] = PDF[ProbabilityDistribution[f[x,\[Mu] ], {x,0, \[Infinity]}]]
\[ScriptCapitalD]= ProbabilityDistribution[f[x,\[Mu] ], {x,0, \[Infinity]}]
Mean[PDF[\[ScriptCapitalD],\[Sigma]]]
TeXForm@HoldForm@Integrate[(2/Pi^2)*(z-\[Sigma])^2/(Sqrt[(1+0.25\[Mu]^2)2*Pi])Exp[-(z-\[Sigma])^2/(1+0.25\[Mu]^2)*Sqrt[(1+0.25\[Mu]^2)*2*Pi]
* Erf[(z-\[Sigma])^2/(1+0.25\[Mu]^2)Sqrt[(1+0.25\[Mu]^2)*2*Pi] ]],{\[Sigma],-Infinity, Infinity}]

Now any Help to compute Variance and skewness and Kurtosis ?

Posted 11 months ago

This question was posted simultaneously on multiple forums. Some answers are posted at Mathematica StackExchange.

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