Editing Normal distribution (section)

=== Fourier transform and characteristic function ===
The [[Fourier transform]] of a normal density {{tmath|f}} with mean {{tmath|\mu}} and variance <math display=inline>\sigma^2</math> is<ref>{{harvtxt |Bryc |1995 |p=23 }}</ref>

<math display=block>
\hat f(t) = \int_{-\infty}^\infty f(x)e^{-itx} \, dx = e^{-i\mu t} e^{- \frac12 (\sigma t)^2}\,,
</math>

where {{tmath|i}} is the [[imaginary unit]]. If the mean <math display=inline>\mu=0</math>, the first factor is 1, and the Fourier transform is, apart from a constant factor, a normal density on the [[frequency domain]], with mean 0 and variance {{tmath|1/\sigma^2}}. In particular, the standard normal distribution {{tmath|\varphi}} is an [[Fourier transform#Eigenfunctions|eigenfunction]] of the Fourier transform.

In probability theory, the Fourier transform of the probability distribution of a real-valued random variable {{tmath|X}} is closely connected to the [[characteristic function (probability theory)|characteristic function]] <math display=inline>\varphi_X(t)</math> of that variable, which is defined as the [[expected value]] of <math display=inline>e^{itX}</math>, as a function of the real variable {{tmath|t}} (the [[frequency]] parameter of the Fourier transform). This definition can be analytically extended to a complex-value variable {{tmath|t}}.<ref>{{harvtxt |Bryc |1995 |p=24 }}</ref> The relation between both is:
<math display=block>\varphi_X(t) = \hat f(-t)\,.</math>