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.
Best Answer
$P(c<Y<d)=\int_c^{d}\int_{-\infty}^{\infty} f_{X,Y}(x,y) dxdy$