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I am trying to find a marginal density of a joint function of $(X,Y)$ where:

$$f_{XY}(x,y) = \frac{\cos(x)}{x + \cos(x)}$$


$$f_Y(y) = \int_{y}^\frac{\pi}{2} \frac{\cos(x)}{x + \cos(x)} dx.$$

I need some help solving this integral.


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I doubt this integral has a simple antiderivative. Your best best is to evaluate the integral numerically at a series of values of $y$, or find an analytical approximation for the integrand that can be integrated. –  Ron Gordon Feb 20 '13 at 21:04
Thanks, that's what I figured. –  user61147 Feb 20 '13 at 21:32
@user61147 Could you be more precise about the domain of $(X,Y)$? –  Sasha Feb 20 '13 at 21:32
@Sasha, This is a problem given in my probability class. The only information I am given about the random variables is that $$0<y<x<\frac{\pi}{2}$$ –  user61147 Mar 5 '13 at 22:21
@user61147 You should confirm correctness of the problem with your instructor. Given $\mathcal{D}= \left\{ (x,y) \colon 0<y<x<\frac{\pi}{2} \right\}$, $$\iint_\mathcal{D} f_{XY}(x,y) \mathrm{d}x\mathrm{d}y \approx 0.3933 \not= 1$$ therefore $f_{XY}$ is not correctly normalized density on the given domain. Coupled with low likelihood for a closed for of the integral you wrote, this hints at a typo. –  Sasha Mar 5 '13 at 22:32

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