DEFINITION.– We say that a real r.v. is Gaussian, of expectation m and of variance σ2 if its law of probability PX:
– admits the density if σ2 ≠ 0 (using a double integral calculation, for example, we can verify that ;
– is the Dirac measure δm if σ2 = 0.
If σ2 ≠ 0, we say that X is a non-degenerate Gaussian r.v.
If σ2 = 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 = σ2. 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 σ2, we will write X ~ N(m,σ2).
Characteristic function of X ~ N(m,σ2)
Let us begin by first determining the characteristic function of X0 ~ 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 ...