Editing Piling-up lemma (section)

== Expected value formulation ==
The piling-up lemma can be expressed more naturally when the random variables take values in <math>\{-1,1\}</math>. If we introduce variables <math>\chi_i = 1 - 2X_i = (-1)^{X_i}</math> (mapping 0 to 1 and 1 to -1) then, by inspection, the XOR-operation transforms to a product:

:<math>\chi_1\chi_2\cdots\chi_n = 1 - 2(X_1 \oplus X_2\oplus\cdots\oplus X_n) = (-1)^{X_1 \oplus X_2\oplus\cdots\oplus X_n}</math>

and since the [[expected value]]s are the imbalances, <math>E(\chi_i)=I(X_i)</math>, the lemma now states:

:<math>E\left(\prod_{i=1}^n \chi_i \right)=\prod_{i=1}^nE(\chi_i)</math>

which is [[Product distribution|a known property of the expected value for independent variables]].

For [[Dependent and independent variables|dependent variables]] the above formulation gains a (positive or negative) [[covariance]] term, thus the lemma does not hold. In fact, since two [[Bernoulli distribution|Bernoulli]] variables are independent if and only if they are uncorrelated (i.e. have zero covariance; see [[Uncorrelatedness (probability theory)|uncorrelatedness]]), we have the converse of the piling up lemma: if it does not hold, the variables are not independent (uncorrelated).