The Exponential and Uniform Distributions

The one idea: The uniform distribution says every value in a range is equally likely — flat, no favorites. The exponential distribution describes the time or distance until the next event in a random arrival process, and it comes with a strange, specific property called memorylessness: having already waited a while gives you zero information about how much longer you'll wait.

⚠️ The trap: "The server hasn't crashed in 40 minutes, and it usually crashes every 30, so it's due any second." If crash times genuinely follow an exponential distribution, that reasoning is backwards. The exponential distribution is memoryless — the probability of waiting at least X more minutes from right now is exactly the same as the probability of waiting X minutes from the very start, no matter how long you've already waited; the clock does not owe you anything for the time already elapsed.

Both distributions in this lesson are continuous — probability is an area under a curve over an interval, never a single point (the continuous distributions lesson covers this mechanic in depth for the normal curve). The discrete distributions lesson covered the Poisson distribution — counts of events over a fixed interval. The exponential distribution is the Poisson's natural partner: where Poisson answers "how many events in this window," exponential answers "how long until the next one."

The uniform distribution: flat probability across a range

The uniform distribution describes a variable equally likely to take any value in [a, b] — no value in the range is more likely than any other. Its density function is a flat line:

f(x) = 1/(b − a) for a…