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"Memoryless" Property

An exponential random variable X has the property that "the future is independent of the past."   That is, the fact "that it hasn't happened yet, tells us nothing about how much longer it will take before it does happen."

Mathematically, this is stated as:

P(T ³ t+t0 | T³t0) = P(T³t) = e-lt.
Interpret T as the time for the next "happening."  It is given that it has been t0 units of time since the last "happening."  The above equation states that the probability that there will be an additional t units of time for the next "happening" is not dependent on the time since the last "happening" (namely t0) - it only depends on t.

WHY ?

By the multiplicative law of probability

P(T³t+t0 | T³t0) × P(T³t0) = P(T³t+t0 AND T³t0) = P(T³t+t0)
Þ P(T³t+t0 | T³t0) × exp(-lt0) = exp[-l(t+t0)] = exp(-lt0)× exp(-lt)
Therefore
P(T³ t+t0 | T³t0) = exp(-lt) = P(T³t).
Again, this property states the following: given that t0 units have elapsed since the last arrival, the probability that an additional t units will elapse before the next arrival is the same as the original probability that t units will elapse until the next arrival.
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Jayant Rajgopal