A variable  is memoryless with respect to  if, for all  with ,

(1)

Equivalently,

			(2)
			(3)

The exponential distribution satisfies

			(4)
			(5)

and therefore

			(6)
			(7)
			(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.