I'm trying to determine a joint PDF for random variables $X$ and $Y$ in this problem:
$(X,Y)$ is uniformly distributed on the subset of $\mathbb R^2$, defined by $0<X<2$ and $0 <Y<X^3$.
I'm not sure I understand what's meant by "uniformly distributed". Are they saying all valid $(X,Y)$ combinations have equal probability? How does one even begin to define a joint PDF for this?
Would it be something like
$f_{X,Y}(x,y) = \begin{cases}k,&0<X<2, 0 <Y<X^3 \\0,& \text{otherwise} \end{cases}$ Where $k$ is some positive constant
It means
$$f_{X,Y}(x,y) = \begin{cases}k,&0<X<2, 0 <Y<X^3 \\0,& \text{otherwise} \end{cases}$$
where $k$ is a positive constant.
To find $k$, we have to find the area of the support.
we have $$k^{-1}= \int_0^2 x^3 \, dx$$