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Given

$$X \sim \text{Beta}(\alpha,\beta)$$ (where $\alpha=\beta$, if that helps) and

$$\theta \sim \text{Uniform}(0, \pi/2).$$

I'm trying to find a formula for $P(Y)$ (or even the cdf) of

$$Y = X + (1-2X)\cos(\theta)$$

on the domain $(0,1)$.

I know from here that given $C = \cos(\theta),$ its PDF is

$$P(C) = \frac{2}{\pi\sqrt{1-C^2}}$$

and of course the pdf of a beta distribution is

$$P(X) = \frac{X^{\alpha-1}(1-X)^{\beta-1}}{\mathrm{Beta}(\alpha, \beta)}.$$

But combining them is getting beyond my skills as an engineer.

EDIT: As it seems a closed form is not possible for this, could someone please show me how to formulate the integral to calculate this sort of PDF? There may be some way I can reformulate the problem to be more solution-friendly if I could wrap my head around how compound distributions of this type are built mathematically.

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  • 2
    $\begingroup$ I can obtain analytic solutions for the PDF when $\alpha=\beta$ are integral, but they are messy and get exponentially messier as $\alpha$ grows, suggesting there won't be any nice formula. Therefore they would be of little use for analysis. If you just need the distribution for particular $\alpha,\beta$ then numerical integration or even simulation will work well. If you need to analyze this distribution to obtain general properties, then you will have to work with the integrals without directly evaluating them. $\endgroup$ – whuber Jan 18 '18 at 15:26
  • $\begingroup$ An integral solution would be worth an answer for me, unless someone else has some brilliant closed form, whether it's for $\alpha = \beta$ or not. I'm also just interested in the process for formulating the integral. $\endgroup$ – Daniel F Jan 19 '18 at 5:57

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