Suppose that a person’s score $X$ on a mathematics aptitude test is a number between $0$ and $1$, and that his score $Y$ on a music aptitude test is also a number between $0$ and $1$. Suppose further that in the population of all college students in the United States, the scores $X$ and $Y$ are distributed according to the following joint p.d.f.:
$f(x,y)= \begin{cases} \frac{2}{5}(2x+3y) & 0 \leq x\leq 1 \text{ and } 0 \leq y\leq 1 \\ 0 & \text{otherwise} \\ \end{cases}$
What proportion of college students obtain a score greater than $0.8$ on the mathematics test?
Solution: $\int_0^1\int_{0.8}^{1}f(x,y)dxdy = 0.264$
I was thinking that to solve this I'd need to find the continuous marginal for $x$, since we're testing for any value of $y$. I was wondering why this method doesn't work.
$f_x(x) = \int_{-\infty}^{\infty}f(x,y)dy=\int_{0}^{1}\frac{2}{5}(2x+3y)dy = \frac{4}{5}x + \frac{3}{5} = f_x(0.8)=\frac{37}{25}$
Note that $f_X(x) = \int_{0}^{1}\frac{2}{5}(2x+3y)dy = \frac{1}{5}(4x+3), \ 0 < x < 1$ is the marginal pdf. You can verify the answer: $P(X>0.8) = \int_{0.8}^1 \frac{1}{5}(4x+3) = 0.264$.
In any case, you just calculate $P(X>0.8)$, using either the joint pdf $f_{X,Y}$ or the marginal pdf $f_X$.