Exponential Distribution

Memorylessness

$\mathbb{P}(X > s + t \mid X > s) = \mathbb{P}(X > t)$ — the past tells you nothing about the future

Arrival timeline

$\lambda$
s

 

Unconditional PDF

Conditional PDF given X > s

Key ideas

  • Memorylessness means the conditional distribution of remaining wait time, given you've already waited s units, is identical to the original distribution. The past gives zero information about the future
  • Only the exponential has this property among continuous distributions (the geometric has it among discrete ones). Any other distribution's conditional PDF changes shape as s increases
  • Both PDFs above are identical regardless of s. This isn't an approximation — it's an exact algebraic property: $f(t \mid X > s) = \lambda e^{-\lambda t}$ for all $s \geq 0$
  • Practical consequence if arrivals follow a Poisson process, knowing that no bus has come in the last 20 minutes doesn't make one more likely to arrive soon. The expected remaining wait is always $\frac{1}{\lambda}$