Editing Normal distribution (section)

=== The Kac–Bernstein theorem ===
The [[Kac–Bernstein theorem]] states that if <math display="inline">X</math> and {{tmath|Y}} are independent and <math display=inline>X + Y</math> and <math display=inline>X - Y</math> are also independent, then both ''X'' and ''Y'' must necessarily have normal distributions.<ref name="Lukacs">{{harvtxt |Lukacs |King |1954 }}</ref><ref>{{cite journal| last1=Quine| first1=M.P. |year=1993|title=On three characterisations of the normal distribution |url=http://www.math.uni.wroc.pl/~pms/publicationsArticle.php?nr=14.2&nrA=8&ppB=257&ppE=263 |journal=Probability and Mathematical Statistics|volume=14 |issue=2 |pages=257–263}}</ref>

More generally, if <math display=inline>X_1, \ldots, X_n</math> are independent random variables, then two distinct linear combinations <math display=inline>\sum{a_kX_k}</math> and <math display=inline>\sum{b_kX_k}</math>will be independent if and only if all <math display=inline>X_k</math> are normal and <math display=inline>\sum{a_kb_k\sigma_k^2=0}</math>, where <math display=inline>\sigma_k^2</math> denotes the variance of <math display=inline>X_k</math>.<ref name="Lukacs" />