A variable is memoryless with respect to if, for all with ,
(1)
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Equivalently,
(2)
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(3)
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The exponential distribution satisfies
(4)
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(5)
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and therefore
(6)
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(7)
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(8)
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is the only memoryless random distribution.
If and are integers, then the geometric distribution is memoryless. However, since there are two types of geometric distribution (one starting at 0 and the other at 1), two types of definition for memoryless are needed in the integer case. If the definition is as above,
(9)
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then the geometric distribution that starts at 1 is memoryless. If the definition becomes
(10)
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then the geometric distribution that starts at 0 is memoryless. Note that these two cases are equivalent in the continuous case.
A useful consequence of the memoryless property is
(11)
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where indicates an expectation value.