Multinomial probabilities and the ϰ 2 and X2distributions

The usual approximation to the logarithm of a multinomial probability correctly gives its asymptotic limit in terms of the X2 distribution (of sums of squares of normal deviates) and the X2 statistic (sum of [observed minus expected]2/expected frequency), but not how it approaches this limit. A formula is derived that appears to be valid under wider conditions; this also throws light on the range of validity of the X2 goodness-of-fit test, and the relation of the log of the likelihood to X2. The expected frequencies can be quite small provided they are nearly equal.