Discrete-Time Markov Chains moved randomness through discrete ticks of a clock. Real arrivals — trades, defaults, customer visits, insurance claims — happen in continuous time, at no fixed tick rate. The exponential distribution is the continuous-time analogue of the coin flip that drove the random walk: it is the unique continuous distribution with the memoryless property, and that single property is what makes the Poisson process (built in the next lesson) tractable in exactly the way the Markov property made discrete chains tractable.
Learning objectives. By the end of this note you should be able to:
- state the exponential distribution's CDF, density, mean, and variance;
- state and prove the memoryless property, and explain why it characterises the exponential distribution among continuous distributions;
- compute the distribution of the minimum of independent exponentials and the identity of the winner of an "exponential race";
- state the sum-of-exponentials-is-Gamma theorem.
In this lesson
- Definition and basic moments
- The memoryless property
- Exponential races
- Sums of exponentials
The full lesson, with worked examples and exercise solutions, is for members.
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