Question
Let $X$ and $Y$ have a joint continuous density function of $$f(x, y) = c\frac y x, \quad 0 < x < 3, \quad 0 \leq y \leq \frac 1 3 x,$$ where $c$ is a constant to be determined.
$(a)\quad$ Determine the value of $c$.
$(b)\quad$ Are $X$ and $Y$ independent? Why or why not?
$(c)\quad$ Find the marginal density function of Y.
My working
$$\begin{aligned}
\int^3_0\int^{\frac 1 3 x}_0 c \frac y x\ \mathrm{d}y\ \mathrm{d}x & = \int^3_0 \frac 1 {18} cx\ \mathrm{d}x
\\[5 mm] & = \frac 1 4 c
\\[5 mm] & = 1
\end{aligned}$$
$$\implies c = 4$$
X and Y are not independent as the range of $y$ depends on $x$.
$$f(y) = \int^3_0 4 \frac y x\ \mathrm{d}x$$
However, this is where I am stuck. The integral above cannot be evaluated at $x = 0$, so I know I must have gone wrong, though I know not where.
Any intuitive explanations will be greatly appreciated š
Edit
As it turns out, I made a simple careless mistake, as pointed out by some really keen eyes!
Best Answer
the correct integral bouds are the following
$$f_Y(y)=4y\int_{3y}^{3}\frac{1}{x}dx$$
here is a drawing of the joint support