$X$ and $Y$ have a joint distribution density function. Working out a marginal density function for $X$ and $Y$

Question

$f_{X,Y}(x,y) = \frac{3}{2}(x^2+y^2)$ if $0 \lt x \lt 1$ and $0 \lt y \lt 1,$ or $0$ otherwise.

I want to find the marginal probability density function of $X$ and $Y$ and then find $Pr(0 \lt x \lt \frac{1}{2}$ and $\frac{1}{2} \lt Y \lt 1)$

The Wikipedia page tells me the marginal probability density function can be written as:

$$p_X(x) = \int_y p_{X,Y} (x,y) dy$$

Does this mean it wants $$p_X(x)= \int_0^1 \frac{3}{2}(x^2+y^2) dy?$$ Can I workout $P_Y(y)$ by interchanging the x and y terms?

Answer 1

The first part is right, the second part you have to integrate $p_{X,Y}$ over the small square. If X and Y are independent then $p_{X,Y}=p_{X} \times p_{Y}$ then what you say would be OK.