Editing Expected value (section)

===Relationship with characteristic function===
The probability density function <math>f_X</math> of a scalar random variable <math>X</math> is related to its [[characteristic function (probability)|characteristic function]] <math>\varphi_X</math> by the inversion formula:
<math display="block">f_X(x) = \frac{1}{2\pi}\int_{\mathbb{R}} e^{-itx}\varphi_X(t) \, dt.</math>

For the expected value of <math>g(X)</math> (where <math>g:{\mathbb R}\to{\mathbb R}</math> is a [[Measurable function|Borel function]]), we can use this inversion formula to obtain
<math display="block">\operatorname{E}[g(X)] = \frac{1}{2\pi} \int_\Reals g(x) \left[ \int_\Reals e^{-itx}\varphi_X(t) \, dt \right] dx.</math>

If <math>\operatorname{E}[g(X)]</math> is finite, changing the order of integration, we get, in accordance with [[Fubini theorem|Fubini–Tonelli theorem]],
<math display="block">\operatorname{E}[g(X)] = \frac{1}{2\pi} \int_\Reals G(t) \varphi_X(t) \, dt,</math>
where
<math display="block">G(t) = \int_\Reals g(x) e^{-itx} \, dx</math>
is the [[Fourier transform]] of <math>g(x).</math> The expression for <math>\operatorname{E}[g(X)]</math> also follows directly from the [[Plancherel theorem]].