The Memoryless Property

Probability · Axiom Academy

LESSON The Memoryless Property A unique characteristic that defines the exponential distribution 1. Definition of Memoryless Property A random variable X has the memoryless property if the conditional probability of exceeding s + t, given that it already exceeds s, equals the unconditional probability of exceeding t. In other words, past waiting has no effect on future waiting. This equation says: "Given I've waited s time units, the probability of waiting at least t more is the same as if I just started waiting." 2. Why Exponential is Memoryless We can verify that the exponential distribution satisfies the memoryless property using its survival function P(X > x) = e^(-λx). The key is showing that conditional and unconditional probabilities are equal. Imagine waiting for a bus where arrivals follow an exponential distribution. If you've already waited 10 minutes, the probability of waiting another 5 minutes is exactly the same as if you just arrived. The distribution has no "memory" of the 10 minutes already spent waiting.

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