Truth be told, I don't really have an issue with this problem in general, but in it's calculation. Let me explain. We need to find $E[Y]$ when $f(x,y) = \frac{x}{3}e^{-xy}$, $1<x<4$ and $y>0$. So, first I calculated the marginal pdf of $y$ by integrating the joint pdf over the entire domain of $x$, and the calculate $E[Y]$ normally. During the calculation, however, at a point I need to calculate the integral $\int_{0}^{\infty} \frac{e^{-4y}}{y} \ dy$. I do not know how to solve for this integral. Previously, I have encountered this kind of integral in other problems too, and I don't know how to solve it. The solutions found on the internet are beyond the scope of my syllabus, so I know something that I am doing is wrong and there is a way to avoid this. So, is there any other approach to solve this sum so that I don't encounter this integral, or even if I do, how do I solve it?
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2$\begingroup$ It looks like you made a mistake. Either rework your answer or consider an alternative approach, such as computing the marginal of $X$ first or even directly doing the double integral. Notice that for $x\gt 0,$ $$\int_0^\infty y e^{-xy}\,\mathrm dy = \Gamma(2)\frac{1}{x}=\frac{1}{x}.$$ $\endgroup$– whuber ♦Commented Mar 21 at 14:11
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