The probability of obtaining an average value z is the product of the probabilities of obtaining arbitrary individual mean values x_i , but with the constraint that the average is z . We can express this through the following expression
\tilde{p}(z)=\int dx_1p(x_1)\int dx_2p(x_2)\dots\int dx_mp(x_m) \delta(z-\frac{x_1+x_2+\dots+x_m}{m}),where the \delta -function enbodies the constraint that the mean is z . All measurements that lead to each individual x_i are expected to be independent, which in turn means that we can express \tilde{p} as the product of individual p(x_i) . The independence assumption is important in the derivation of the central limit theorem.