Memoryless

A memoryless random variable is a nonnegative random variable satisfying, for all with ,

(1)

Equivalently,

			(2)
			(3)

For , the exponential distribution satisfies

			(4)
			(5)

and therefore

			(6)
			(7)
			(8)

so the exponential distribution is memoryless. It is the only nondegenerate continuous distribution on the nonnegative real numbers with this property.

For nonnegative integers and , the geometric distribution has the corresponding discrete memoryless property. 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.

For a random variable with finite mean satisfying the strict-inequality definition, a useful consequence is

(11)

where indicates an expectation value.