# memoryless random variable

A non-negative-valued random variable^{} $X$ is *memoryless* if
$P(X>s+t\mid X>s)=P(X>t)$ for $s,t\ge 0$.

In words, given that a certain event did not occur during time period $s$ *in the past*, the chance that an event will occur after an additional time period $t$ *in the future* is the same as the chance that the event would occur after a time period $t$ from the beginning, regardless of how long or how short the time period $s$ is; the memory is *erased*.

From the definition, we see that

$$P(X>t)=P(X>s+t\mid X>s)=\frac{P(X>s+t\text{and}Xs)}{P(Xs)}=\frac{P(Xs+t)}{P(Xs)},$$ |

so $P(X>s+t)=P(X>s)P(X>t)$ iff $X$ is memoryless.

An example of a discrete memoryless random variable is the geometric random variable^{}, since $P(X>s+t)={(1-p)}^{s+t}={(1-p)}^{s}{(1-p)}^{t}=P(X>s)P(X>t)$, where $p$ is the probability of $X$=success. The exponential random variable is an example of a continuous memoryless random variable, which can be proved similarly with $1-p$ replaced by ${e}^{-\lambda}$. In fact, the exponential random variable is the only continuous random variable having the memoryless property.

Title | memoryless random variable |
---|---|

Canonical name | MemorylessRandomVariable |

Date of creation | 2013-03-22 14:39:49 |

Last modified on | 2013-03-22 14:39:49 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 8 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 60K05 |

Classification | msc 60G07 |

Related topic | MarkovChain |