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$X$ and $Y$ are jointly distributed on the region $ 0 \leq y \leq x, y \leq 1, x \leq 2$ The joint distribution has a constant density over the entire region. Find the marginal densities of $X$ and $Y$.

The joint density is $ f(x,y) = \frac{2}{3}$. What I don't understand is why $$ f_Y(y) = \int_y^2\frac{2}{3}\, \mathrm{d}x = \frac{2}{3}(2-y).$$

Why is the lower bound of the integral set to $y$?

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1 Answer 1

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Because $X$ and $Y$ are jointly distributed on a region for which $X$ is always at least as large as $Y$.

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