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Mathsey
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Homework Statement
So I have an issue evaluating the integral for a joint probability distribution given by:
[tex]Pr(R) = \displaystyle \int_{0}^{r_{max}}\int_0^{2\pi}\int_0^{\pi}\sin\theta \delta(R-r\sin\theta)d\theta d\phi dr[/tex]
where I know the relationship between r and R is given by [tex]R=r\sin\theta[/tex]
Are there any special properties of the delta function I should be aware of besides it's sifting property?
Homework Equations
The Attempt at a Solution
I have tried evaluating this by re-writing the integral as
[tex]\int_{0}^{r_{max}}\int_0^{2\pi}\int_0^{\pi}\sin\theta \delta(\sin^{-1}\left(\frac{R}{r}\right) - \theta)d\theta d\phi dr[/tex]
so that it becomes
[tex]\int_{0}^{r_{max}}\int_0^{2\pi} \sin[\sin^{-1}\left(\frac{R}{r}\right)] d\phi dr[/tex]
etc...
but this cannot be evaluated so I think this is just wrong, any help would be great.
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