I have the following PDF: $$f(x)=xe^{-x}$$ for x>0. I am to find the characteristic equation for this PDF. I have so far: $$\phi(u)=\int_{0}^{\infty}xe^{-x+iux}dx= \int_{0}^{\infty}xe^{x(iu-1)}dx $$ Through integration by parts I now have $$\phi(u)=\frac{xe^{x(iu-1)}}{iu-1}\Big|_0^{\infty}-\frac{e^{x(iu-1)}}{(iu-1)^2}\Big|_0^{\infty}$$ Since the integral contains imaginary parts I am confused as to how to continue solving. Any help would be greatly appreciated.
2026-03-31 23:25:14.1774999514
Finding a characteristic function for a Gamma random variable
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