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Date: 24-3-2021
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A variable is memoryless with respect to if, for all with ,
(1) |
Equivalently,
(2) |
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(3) |
The exponential distribution satisfies
(4) |
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(5) |
and therefore
(6) |
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(7) |
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(8) |
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) |
then the geometric distribution that starts at 1 is memoryless. If the definition becomes
(10) |
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) |
where indicates an expectation value.
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