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Given that the pdf of $X+Y$ is the convolution of pdfs $X$ and $Y$; show that $M_{X+Y}$ is $M_XM_Y$ where $M$ is the moment generating function. $X and Y$ are independent and continuous. I am confused how to proceed form here(see picture). Thank you.

My approach: enter image description here

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Your last line should have been $$m_{X+Y}(t) = \int_{-\infty}^\infty e^{ts} f_{X+Y}(s) \mathop{ds}= \int_{-\infty}^\infty e^{ts} \int_{-\infty}^\infty f_X(s-y) f_Y(y) \mathop{dy} \mathop{ds}.$$ Making the change of variables $s=x+y$ gives you the answer.

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