We have $f_{X,Y} = cxy$ when $0 \le x \le y \le 1$ (and $f_{X,Y} = 0$ otherwise). $c$ is a normalization constant.
I'm trying to compute the marginal PDF of $Y$ from the above.
To find the marginal PDF of $Y$, we're going to integrate the joint PDF over $x$, but I'm not sure over which range we're integrating?
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Is the integration range for the computation $[0, y]$ (the values $x$ can take) or is it $[0,1]$?
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What is the marginal PDF of $Y$?
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Over which range is the resulting marginal PDF of $Y$ defined?
Best Answer
$Y$ can realise values between $0$ and $1$.
When $Y$ has a particular value of $y$, then $X$ shall have some value $x$ where $0\leqslant x\leqslant y$.
Thus:
$$\begin{align}f_{\small Y}(y)&=\int_\Bbb R c x y\,\mathbf 1_{0\leqslant x\leqslant y\leqslant 1}\,\mathrm d x\\[1ex]&= cy\,\mathbf 1_{0\leqslant y\leqslant 1}\int_0^y x\,\mathrm d x\end{align}$$