The memoryless property (also called the forgetfulness property) means that a given probability distribution is independent of its history. Any time may be marked down as time zero.

If a probability distribution has the memoryless property the likelihood of something happening in the future has no relation to whether or not it has happened in the past. The history of the function is irrelevant to the future.

## Technical Definition of the Memoryless Property

A discrete random variable *X* is memoryless with respect to a variable *a* if, (for positive integers *a* and *b*) the probability that *X* is greater than *a + b* given that *X* is greater than *a* is simply the probability of *X* being greater than *b*. Symbolically, we write:

P( X > a + b | x > a ) = P ( x > b )

To make this specific, let *a* be 5 and *b* 10. If our probability distribution is memoryless, the probability *X* > 15 if we know *X* > 5 is exactly the same as the probability of *X* being greater than ten.

Note that this is not the same as the probability of *X* being greater than 15, as it would be if the events *X* > 15 and *X* > 5 were independent.

We say that a continuous random variable *X* (over the range of reals) is memoryless if for every real *s, t*

P( X > t + s | x > t ) = P ( x > s ).

## Examples of the Memoryless Property

Tossing a fair coin is an example of probability distribution that is memoryless. Every time you toss the coin, you have a 50 percent chance of it coming up heads. It doesn’t matter whether or not the last five times you threw the dice it came up consistently tails; the probability of heads in the next throw is always going to be zero.

For a real life example, consider independent failures of computer hardware. When figuring the probability of a (new, independent) hardware failure it doesn’t matter how frequently or when your hardware failed in the past. The probability it will fail five minutes from now is independent of the fact that it hasn’t failed for three months. The probability distribution can be modeled by the exponential distribution or Weibull distribution, and it’s memoryless.

In fact, the only continuous probability distributions that are memoryless are the exponential distributions. If a continuous *X* has the memoryless property (over the set of reals) *X* is necessarily an exponential.

The discrete geometric distribution (the distribution for which P(X =n)=p(1−p)^{n−1}, for all n≥1) is also memoryless.

## Sources:

Conditional Probabilities and the Memoryless Property

Wolfram Memoryless

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