I am trying to derive the distribution for the sum of two random vectors, such that:(adsbygoogle = window.adsbygoogle || []).push({});

[tex]

\begin{align}

X &= L_1 \cos \Theta_1 + L_2 \cos \Theta_2 \\

Y &= L_1 \sin \Theta_1 + L_2 \sin \Theta_2

\end{align}

[/tex]

With:

[tex]

\begin{align}

L_1 &\sim \mathcal{U}(0,m_1) \\

L_2 &\sim \mathcal{U}(0,m_2) \\

\Theta_1 &\sim \mathcal{U}(0, 2 \pi) \\

\Theta_2 &\sim \mathcal{U}(0, 2 \pi)

\end{align}

[/tex]

In other words, two vectors, each with a uniformly random direction, and each with a magnitude uniformly random between zero and [itex]m_1[/itex] or [itex]m_2[/itex], respectively. Is this even worth trying to calculate analytically?

I've tried to break the problem down into simpler parts. First, I calculated the PDF of [itex]S_1 = \cos \Theta_1[/itex] as:

[tex]

f_{S_1}(s_1) = \frac{1}{\pi \sqrt{1 - {s_1}^2}}

[/tex]

Then I thought, if we ignore [itex]L_1[/itex] and [itex]L_2[/itex], how can I find the PDF of [itex]S_1 + S_2 = \cos \Theta_1 + \cos \Theta_2[/itex]? I thought I could try multiplying the characteristic functions of [itex]S_1[/itex] and [itex]S_2[/itex], so I tried taking the Fourier transform of [itex]f_{S_1}(s_1)[/itex] in both MATLAB and Mathematica, but MATLAB just choked on it, and Mathematica returns something involving the Henkel function which looks too complex to use.

On Wikipedia I found something called the Arcsine distribution, which has a CDF similar to [itex]F_{S_1}[/itex]. This is a special case of the Beta distribution, which Wikipedia does give the characteristic function for, but I'm not sure I can use it given that the CDF for the Arcsine distribution is slightly different than mine. However, this leads me to believe that the characteristic function for [itex]S_1[/itex] is tractable.

I really don't know anything about probability, I'm just reading Wikipedia and trying to make some sense of this problem. I would really appreciate someone telling me where to look next, or at least that what I'm trying to do is analytically impossible!

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# PDF of sum of random vectors

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