Given the joint density function f of X and Y, find the density of X:
$$
f(x, y) =
\begin{cases}
xe^{-x + y} &\quad x>0, y>0 \\
0, &\quad \text{otherwise}
\end{cases}
$$
My approach to computing the marginal is the following:
$$
f_X = \int_{-\infty}^{\infty} f(x, y) dy =
\begin{cases}
\int_0^{\infty} f(x, y) dy = xe^{-x}\int_0^{\infty} e^y dy = \infty \textbf{???!!!}, &\quad x > 0 \\
0, &\quad \text{otherwise}
\end{cases}
$$
My problem is that the pdf goes to $\infty$ if x>0. Am I correctly computing the pdf and the integral, or am I missing something?
Best Answer
I suspect the joint density should have been $$ f(x, y) = \begin{cases} xe^{-(x + y)} &\quad x>0, y>0 \\ 0, &\quad \text{otherwise} \end{cases} $$ since this integrates to $1$, i.e. the author probably forgot the parentheses.
Your approach to computing the marginal density w.r.t. $X$ is fine.