Expected value

===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>f_X(x) = \frac{1}{2\pi}\int_{\mathbb{R}} e^{-itx}\varphi_X(t) \, \mathrm{d}t.</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>\operatorname{E}[g(X)] = \frac{1}{2\pi} \int_{\mathbb R} g(x)\left[ \int_{\mathbb R} e^{-itx}\varphi_X(t) \, \mathrm{d}t \right]\,\mathrm{d}x.</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>\operatorname{E}[g(X)] = \frac{1}{2\pi} \int_{\mathbb R} G(t) \varphi_X(t) \, \mathrm{d}t,</math>
where
:<math>G(t) = \int_{\mathbb R} g(x) e^{-itx} \, \mathrm{d}x</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]].