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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?

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  • $\begingroup$ There must be a mistrake in your formula, since $f(x,y)$ cannot be a density function, since its double integral isn't 1. $\endgroup$ Commented Mar 19, 2021 at 9:05

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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.

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