DEFINITION.– We say that a real r.v. is Gaussian, of expectation *m* and of variance σ^{2} if its law of probability *P _{X}*:

– admits the density if σ^{2} ≠ 0 (using a double integral calculation, for example, we can verify that ;

– is the Dirac measure *δ _{m}* if σ

If *σ*^{2} ≠ 0, we say that *X* is a non-degenerate Gaussian r.v.

If *σ*^{2} = 0, we say that *X* is a degenerate Gaussian r.v.; *X* is in this case a “certain r.v.” taking the value *m* with the probability 1.

*EX = m, Var X = σ*^{2}. This can be verified easily by using the probability distribution function.

As we have already observed, in order to specify that an r.v. *X* is Gaussian of expectation *m* and of variance *σ*^{2}, we will write *X ~ N*(*m,σ*^{2}).

*Characteristic function of**X ~ N*(*m,σ*^{2})

Let us begin by first determining the characteristic function of *X _{0} ~ N*(0,1):

We can easily see that the theorem of derivation under the sum sign can be applied:

Following this, with integration by parts:

The resolution of the differential ...

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