we are trying to throw up warnings for when customers buying behavior might have changed. Our current indicator is the number of purchases in a given time period and we believe that this process satisfies the condition for assuming a Poisson distribution.
The idea is that when the likelihood for the exact number of purchases (probability mass function) in the current timeframe is lower than x%, we throw a warning and somebody should take a look at this.
If the number of purchases gets too large and technical limitations make it impossible to calculate the Poisson distribution, we are estimating it with a normal distribution with SD = sqrt(number of purchases in current timeframe).
We are getting results though, which seem intuitively wrong. Example: 229 purchases last year, 225 this year. Intuitively I would say, that this is close enough to discard as normal fluctuation. But the likelihood for this result is (appr. with normal dist.) just 2,55%.
Questions: Am I right in thinking that the cumulative distribution function would be more appropriate to use? If so, why? My line of thinking would be that we have one result and we want the likelihood for that result not the likelihood of that result and all that are worse.