information:filtering
Relationship with characteristic function

The probability density function f X {\displaystyle f_{X}} of a scalar random variable X {\displaystyle X} is related to its characteristic function φ X {\displaystyle \varphi _{X}} by the inversion formula: f X (x) = 1 2 π ∫ R e − i t x φ X (t) d t. {\displaystyle f_{X}(x)={\frac {1}{2\pi }}\int _{\mathbb {R} }e^{-itx}\varphi _{X}(t)\,dt.}

For the expected value of g (X) {\displaystyle g(X)} (where g: R → R {\displaystyle g:{\mathbb {R} }\to {\mathbb {R} }} is a Borel function), we can use this inversion formula to obtain E = 1 2 π ∫ R g (x) d x. {\displaystyle \operatorname {E} ={\frac {1}{2\pi }}\int _{\mathbb {R} }g(x)\leftdx.}

If E {\displaystyle \operatorname {E} } is finite, changing the order of integration, we get, in accordance with Fubini–Tonelli theorem, E = 1 2 π ∫ R G (t) φ X (t) d t, {\displaystyle \operatorname {E} ={\frac {1}{2\pi }}\int _{\mathbb {R} }G(t)\varphi _{X}(t)\,dt,} where G (t) = ∫ R g (x) e − i t x d x {\displaystyle G(t)=\int _{\mathbb {R} }g(x)e^{-itx}\,dx} is the Fourier transform of g (x). {\displaystyle g(x).} The expression for E {\displaystyle \operatorname {E} } also follows directly from the Plancherel theorem.