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Let $X_1$ and $X_2$ be two independent exponential random variables with $\lambda_1 \neq \lambda_2$ as parameters. I want to find the density function of $ Z = aX_1 + bX_2 $ with $a,b > 0$.

This is what I have done so far:

$$ f_Z(z) = \int_{-\infty}^{+\infty}f_{X_1}(z-x_2)f_{X_2}(x_2) dx_2 $$

$$ = \int_0^z a\lambda_1 e^{-a\lambda_1(z-x_2)}b\lambda_2 e^{-b\lambda_2x_2}dx_2 $$ and from here calculations. I just wanted to make sure that this procedure is the right one.

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There's one mistake you're making. Bear in mind $bX_2\sim\operatorname{Exp}(\lambda_2/b)$, so as to scale the mean from $1/\lambda_2$ to $b/\lambda_2$. Thus each appearance of $a,\,b$ in your integral should be replaced with $1/a,\,1/b$. Apart from that, though, you're on the right track.

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  • $\begingroup$ I could sense there was something wrong. Thank you! $\endgroup$ – qcc101 Nov 5 '18 at 20:04
  • $\begingroup$ @qcc101 No problem. You could also have spotted it by using dimensional analysis. $\endgroup$ – J.G. Nov 5 '18 at 20:08

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