Question: If the joint frequency function of random variables $X$ and $Y$ is given by:

$f\_{X,Y}(x,y)=\alpha \beta e^{-\alpha x-\beta y}$ $\,\,\,\,x\geq 0$ $\,\,\,\,y\geq 0$

then to get, for example, the marginal density function of $Y$ we would integrate the joint frequency function with respect to $X$.

This would give $f\_Y(y)=\beta e^{-\beta y}$.

What I'm not totally clear on is how you would use this. I understand the discrete case, where if you want the marginal density of $Y=2$, you sum the row or column of $Y=2$ across all the $X$ values.

Here, how would you get the marginal probability that $Y$ takes on a value in a certain interval? It seems like you should integrate but what would the bounds be? Any help is appreciated.

Answer: 
the given density can be rewritten in the following way

$$f\_{XY}(x,y)=\alpha e^{-\alpha x}\times \beta e^{-\beta x}=f\_X(x)\times f\_Y(y)$$

thus you immediately get your marginals as independent exponentials
The marginal density is defined as $$f\_Y(y) = \int f(x, y) dx$$. In this case, you treat $y$ as a constant, giving you $$\int\_0^\infty \alpha \beta e^{-\alpha x - \beta y} dx = \beta e^{-\beta y} \int\_0^\infty \alpha e^{-\alpha x} dx = \beta e^{-\beta y}$$