Random variables ξ, η and ζ are pairwise uncorrelated. It means that E(ξ*ζ) = E(ξ)*E(ζ), etc. Is it true that in this case E(ξηζ) = EξEηEζ ? How it can be proven?

Note: we don't know if they are dependent or not, and we don't want to prove it, only correlation.


Toss two coins, set

$\xi = 1$ if the first coin is a head, 0 otherwise,

$\eta = 1$ if the second coin is a head, 0 otherwise,

$\zeta = 1$ if the coins show different symbols, 0 if they are the same.

Then $\xi$, $\eta$ and $\zeta$ are pairwise independent (hence pairwise uncorrelated) random variables with $E(\xi) = E(\eta) = E(\zeta) = \frac 12$ but $\xi\eta\zeta = 0$ with probability one.


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