Continuous distributions: uniform, exponential, power law
The bell is not the only shape of continuous data. Waiting times, money, file sizes, session durations are built differently. Let's cover three more important shapes and learn the telltale signs of each in real data. "Pour" a sample โ the histogram settles onto the density.
PDF โ density: the height itself is not a probability; the probability of an interval = the area under the curve over it. Total area = 1.
All values in [a, b] are equally likely โ the density is a flat "shelf" with no distinguished center. Parameters a and b set the bounds. The model of complete ignorance: a random arrival within an hour, a random number generator, "somewhere between a and b, that's all". Press "pour" โ the real sample\'s histogram approaches the theoretical density.
The list at the top offers the continuous shapes. Start with the uniform distribution (selected by default): every value in the range is equally likely, the density is a flat "shelf". It is the model of complete ignorance: a random arrival within an hour, a random number generator, "somewhere between a and b, that's all we know". Drag the bounds a and b.
Shape recognition is the first practical skill. Spotted a long right tail in revenue or response time? The mean will lie โ take the median and percentiles (p95, p99). A direct continuation of the skew lesson.
Exponential and Poisson are the backbone of queueing and load models: knowing the arrival intensity you can estimate both the average wait and the overload probability. And the uniform underpins simulations and A/A checks, where "honest" randomness is needed.
Time to equipment failure, between server requests, between customer arrivals โ classic exponential quantities on which SLAs and capacity plans are built.
Power laws are everywhere in "social" data: city sizes, incomes, product popularity, follower counts. Everywhere a handful of giants and a long tail of small fry โ and everywhere the mean misleads.