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I want to find the PDF $f_\boldsymbol{X}(\vec{x})$ of a random variable $\boldsymbol{X}\in\mathbb{R}^3$ with positivly normally distributed length $|\boldsymbol{X}|=|x|$ where $x\sim\mathcal{N}(0,\sigma^2)$ and a unit direction (in the unit sphere $S^3$), i.e. $\hat{\boldsymbol{X}} = \boldsymbol{X}|\boldsymbol{X}|^{-1}\sim\mathcal{U}(S^3)$.

Since $\boldsymbol{X}$ results from the trivial composition $\boldsymbol{X}=|\boldsymbol{X}|\hat{\boldsymbol{X}}$, it should be possible to construct $f_\boldsymbol{X}$ from the normal and uniform distribution. A similar approach is used for for different compositions and PDFs, e.g. here or here.

My attempt was the following: Let $\omega\in\mathbb{R}^3$. We know that $$ Pr(\hat{\boldsymbol{X}}\in\omega) = \frac{|\omega ~\cap~S^3|}{|S^3|} $$ and thus $$ Pr(\boldsymbol{X}\in\omega) = Pr(\hat{\boldsymbol{X}} \in \omega|\boldsymbol{X}|^{-1}). $$ Now my most intuitive next step is $$ Pr(\hat{\boldsymbol{X}} \in \omega|\boldsymbol{X}|^{-1}) =\frac{|\omega |\boldsymbol{X}|^{-1}~\cap~S^3|}{|S^3|} $$ which clearly is non-sense, since the random variable $|\boldsymbol{X}|^{-1}$ apprears in the CDF.

How do I go on from here? I would very much like to solve this using first principles and bare probabilities.

zufall
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  • Did you mean for the length of $X$ to be normal with zero mean? I believe it should be strictly positive. $$$$ For the uniform direction part, check out https://mathworld.wolfram.com/SpherePointPicking.html – Gábor Pálovics May 17 '21 at 10:49
  • @GáborPálovics: Your right. I edited it. – zufall May 17 '21 at 10:59

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Let $$X = R \begin{bmatrix} \sin\theta \cos\phi \\ \sin\theta \sin\phi \\ \cos \phi \end{bmatrix}$$ Then if you choose $U, V \sim Uni(0, 1)$ i.i.d, you can get a uniform distribuiton on the sphere R, by using: $$\theta = 2 \pi u$$ $$\phi = \arccos (2v - 1)$$

So what you want is $$U, V \sim Uni(0, 1)$$ $$R \sim |N(0, \sigma^2)|$$ $$\theta = 2 \pi u$$ $$\phi = \arccos (2v - 1)$$

For more informationon this, see: https://mathworld.wolfram.com/SpherePointPicking.html

Gábor Pálovics
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  • As I pointed out in my question, I would like to understand how to get the PDF from the composite PDFs of lengths and direction, in the way the provided likes do. – zufall May 17 '21 at 11:41
  • Your example is different from what you referenced. In the referenced examples, we know the exact relationship between $X$ and $Y$. Here this relationship is not so trivial. You can arrive to a setting with well-defined relationship by using the equations above. Then you can calculte the joint and marginal pdfs with the same methodology. – Gábor Pálovics May 17 '21 at 11:54
  • You're right! The composition into length and direction leads to a dead end, I guess. Thank you! – zufall May 17 '21 at 16:00