Continuous Distributions

Symmetric, bell-shaped, characterized by mean $mu$ and standard deviation $sigma$. The most important distribution in statistics, for two reasons:

1. The Central Limit Theorem: The distribution of sample means approaches normal as sample size grows, regardless of the underlying distribution. This is why so many statistical tests assume normality — in the limit, it's valid even when the raw data isn't normal.
2. Natural prevalence: Heights, measurement errors, sums of many small independent effects — these all tend toward normal in practice.

When it shows up: Test scores. Measurement error. Model residuals (when assumptions hold). Many parametric statistical tests assume normal data.

Equal probability for all values within a specified range [a, b]. The flattest distribution — no value is more likely than any other.

When it shows up: Random number generation. A/B test bucketing. Monte Carlo simulation. Any situation where you have no reason to prefer one value over another.

Models the time between events in a Poisson process. Single parameter: the rate $lambda$ (higher rate = shorter typical waiting time). It has the memoryless property: knowing that you've already waited 5 minutes tells you nothing about how much longer you'll wait.

When it shows up: Time between bus arrivals. Time until a customer churns. Time to component failure. Survival analysis. Any "how long until the next event?" question where events arrive at a steady rate.