Artificial intelligent assistant

Integral of characteristic and density function Let X and Y be random variables (real valued) with density functions $f_X, f_Y$ and characteristic functions $I_X, I_Y$. How can we show that: $ \int_{-\infty}^{\infty}{I_X(y) f_Y(y) e^{-ity}dy} = \int_{-\infty}^{\infty}{I_Y{(x-t)f_X(x)}dx}$ ?

**Hint** : Write out $I_X(y)$ in terms of an integral using that $X$ has density $f_X$. Then apply Fubini's theorem.

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**Edit** : The characeristic function $I_X$ can be written as $$ I_X(t)={\rm E}\left[e^{itX}\right]=\int_{-\infty}^\infty e^{itx}f_X(x)\,\mathrm dx $$ and plugging this into the left-hand side, we get $$ \int_{-\infty}^\infty I_X(y)f_Y(y)e^{-ity}\mathrm dy=\int_{-\infty}^\infty\int_{-\infty}^\infty e^{iyx}f_X(x)e^{-ity}f_Y(y) \,\mathrm dx\,\mathrm dy. $$

Now apply (the complex version of) Fubini's theorem.

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