Let $f: A \to \mathbb{R}, \: A=\left\{(x, y) \in R^{2} | 0<x<1,0<y<x^{2}\right\}$ be a PMF for the joint r.v. $(X,Y)$ given by $f(x,y)=3$. Find marginal distribution of $Y$.
Let be denote the marginal distribution of $Y$ as $f_Y$. This is how I will find it: by calculating the following integral: $f_Y(y)=\int_0^{x^2}3dx$. This is apparently wrong. I can't figure out why?
The joint density of $(X,Y)$ is $$ f_{X,Y}(x,y) = 3\cdot\mathsf 1_A, $$ so to find the marginal density of $Y$ we integrate over all values of $X$. Hence $$ f_Y(y) = \int_{\sqrt y}^1 3\ \mathsf dx = 3 \left(1-\sqrt{y}\right)\mathsf 1_{(0,1)}(y). $$