Exponential distribution

The exponential distribution models the waiting time until the next event in a Poisson process — a radioactive decay, a web server request, a bus arrival. Its rate parameter $λ$ sets how quickly events happen on average.

f (x; λ) = λ e^{- λ x}

mean = 1.00 median = 0.69

λ (rate) 1.0 n (samples) 500

PDF

f (x; λ) = λ e^{- λ x}

. Bars: 500 samples. Mean =

1/ λ

, median =

ln 2/ λ

What to notice

Higher $λ$ shifts mass toward zero — events happen sooner. The distribution compresses toward the left.
Mean vs median. Because the distribution is right-skewed, the mean (= $1/ λ$ ) always exceeds the median (= $ln 2/ λ$ ). Most waiting times are shorter than average.
Heavy right tail. Even with high $λ$ , long waiting times have nonzero probability. Rare but very long waits always exist.

The memoryless property

The exponential is the only continuous distribution with this property:

P (X > s + t ∣ X > s) = P (X > t)

If you’ve already waited s minutes for the bus, the remaining wait has the same distribution as if you had just arrived. The bus has no memory of how long you’ve been there.

This makes the exponential both analytically tractable and sometimes an unrealistic model: many real-world waits do have memory (a server that’s been idle tends to receive a request soon).