Uniform and Exponential Distributions

These two continuous distributions describe very different situations. The uniform distribution treats every value in a range as equally likely, while the exponential distribution describes the waiting time until the next event.

Uniform Distribution

The density is flat, so the probability of landing in an interval is its length divided by the total length: P(c < X < d) = (d − c) / (b − a).

Example: A bus arrives at a random moment within a 20-minute window, so the wait is uniform on [0, 20]. The probability of waiting less than 5 minutes is 5/20 = 0.25, and the average wait is (0 + 20)/2 = 10 minutes.

Exponential Distribution

P(X > x) = e⁻λˣ

• λ (lambda): the average number of events per unit of time.
• x: the amount of time that passes.
• P(X > x): the probability that the wait for the next event lasts longer than x.

If the number of events in an interval follows a Poisson distribution with rate λ, the time between consecutive events follows an exponential distribution with the same λ.

The distribution is also memoryless, meaning the time already waited does not change the probability distribution of the remaining time.

Example: Customer calls arrive at an average rate of 2 per minute. The average gap between calls is 1 / 2 = 0.5 minutes. The probability that the next call takes more than 1 minute is e⁻² ≈ 0.1353, or about 13.5%.