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Let the joint distribution of a random vector be $$f(x,y)=\begin{cases}1,\enspace 0<x<1, x<y<x+1&\\0, \enspace {\rm otherwise}\end{cases}$$

The marginal pdf of $X$ is $Uni(0,1)$, but what is the distribution of $Y$? By definition it should be $$f_Y(y)=\int_0^1 f(x,y){\rm d}x=1$$

but this is not correct. Integration is done over the bounds of $X$, and when $x\in(0,1)$ then $f(x,y)=1$. Is this not correct?

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You have to take into consideration the inequalities $0<x<1, x<y<x+1$ completely.

$$f_Y(y)=\int_{0}^{y}f(x,y)dx=y$$ for $0<y<1$

and $$f_Y(y)=\int_{y-1}^{1}f(x,y)dx=2-y$$ for $$1<y<2.$$

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