Information divergence is more χ2-distributed than the χ2-statistics

For testing goodness of fit it is very popular to use either the χ²-statistic or G²-statistics (information divergence). Asymptotically both are χ²-distributed so an obvious question is which of the two statistics that has a distribution that is closest to the χ²-distribution. Surprisingly, when there is only one degree of freedom it seems like the distribution of information divergence is much better approximated by a χ²-distribution than the χ²-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.