I am having a huge problem with a conceptual problem that I came up with.
Say a company has a distribution that is highly skewed. Something similar to an exponential or lognormal only more extreme. Now pretend the distribution is so skewed that the mean of the distribution is higher than the 99% Percentile of the distribution. (Aka 1-2 extreme higher values caused the mean to be extremely high compared to the rest of distribution).
By definition, if this distribution was used to forecast a future value (aka a random sample from the distribution) would it be true that mean would not be in the 95% Prediction interval?
In my brain, a 95% predition interval is a range that 95% of all future values will fall between. For any distribution this should exactly equal the .025 Percentile on the lower bound, and the .975 percentile on the upper bound... If the mean is higher than the .975 Percentile, then the mean would not be within the '95% prediction interval'.
Am I thinking of this incorrectly? It seems strange to report a forecast as
- Mean Forecasted Value: 6,000,0000
- 95% Prediction Interval: [400,5000].