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We know that if

x \[Distributed] ExponentialDistribution[λ]

then

y = Sqrt[x] \[Distributed] RayleighDistribution[1/Sqrt[2*λ]]

Which we can justify by the following:

Dist = RayleighDistribution[1/Sqrt[2*λ]];
SDist = TransformedDistribution[x^2, x \[Distributed] Dist]

On the other hand, we also know that if

r \[Distributed] RiceDistribution[0, σ]

then

r \[Distributed] RayleighDistribution[σ]

But by the following code, I do not get an Exponential Distribution. Why?

Dist = RiceDistribution[0, 1/Sqrt[2*λ]];
SDist = TransformedDistribution[x^2, x \[Distributed] Dist]
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Dist = RiceDistribution[0, 1/Sqrt[2*λ]];

SDist = TransformedDistribution[x^2, x \[Distributed] Dist];

Compare the PDFs for SDist and ExponentialDistribution

PDF[SDist, x] // FullSimplify

enter image description here

PDF[ExponentialDistribution[λ], x]

enter image description here

While functionally equivalent, the PDFs differ (a difference that is negligible since it has measure zero) is that the PDF for SDist is not defined for x==0. This difference arises because the PDF for the RiceDistribution is not defined for x==0

PDF[Dist, x]

enter image description here

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