Question

`\rm Give \: a \: joint \: density \: function \: of \\ \rm random \: variables \: X \: and \: Y \: as \: follows \\ \small\begin{gathered} \rm{f(x,y) = \begin{cases} \rm x + y, \: \: \: \: \: 0 \leq x \leq1 , \: \: \: \: \: 0 \leq y \leq2 \\ \: \: \: \: \: \: \: \: 0, \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \rm{otherwise}\end{cases}}\end{gathered} \\ \rm Find \: the \: marginal \: density \: of \: X​​` ​

Answer 1

Normally, you would just integrate the joint density over all possible values of Y:

`f_X(x) = \displaystyle \int_(-\infty)^\infty f_(X,Y)(x,y) \, dy`

However, the given joint density is not actually a valid density function, since

`\displaystyle \int_(-\infty)^\infty f_(X,Y)(x,y) \, dx\, dy = \int_0^2 \int_0^1 (x+y) \, dx \, dy = 3 \\eq 1`

so the question is not well-posed.