  For testing goodness of fit it is very popular to use either the chi square
statistic or G statistics (information divergence). Asymptotically both are chi
square distributed so an obvious question is which of the two statistics that
has a distribution that is closest to the chi square distribution.
Surprisingly, when there is only one degree of freedom it seems like the
distribution of information divergence is much better approximated by a chi
square distribution than the chi square statistic. For random variables we
introduce a new transformation that transform several important distributions
into new random variables that are almost Gaussian. For the binomial
distributions and the Poisson distributions we formulate a general conjecture
about how close their transform are to the Gaussian. The conjecture is proved
for Poisson distributions.
