Memorylessness of Distributions

Memorylessness is a property some statistical distributions have. If the future probability of an event only depend on its remaining time, not the time already passed, it has the memorylessness property.

Mathematical Definition

A continuous random variable $X$ (representing time or lifetime) is memoryless if, for any non-negative real numbers $s$ and $t$:

$$P(X > s + t \mid X > s) = P(X > t)$$

What This Means:

• $X > s$: Given that the event hasn't happened in the first $s$ units of time.
• $X > s + t$: The probability that the event takes more than $s + t$ total units of time.
• $P(X > t)$: Is exactly the same as starting fresh from time $0$ and waiting for $t$ units of time.

Examples

1. Memoryless Example: Radioactive Decay or Electronic Components Imagine a lightbulb whose failure model is memoryless. The probability that the bulb lasts for 100 hours more, given it has already lasted for 1,000 hours, is identical to the probability that a brand-new bulb lasts for 100 hours. It does not "wear out" or age over time.

2. NOT Memoryless Example: Human Lifespan or Mechanical Wear A 90-year-old human reaching age 100 has a vastly different probability than a newborn baby reaching age 10. Humans age, accumulate damage, and become more likely to die as time passes. Thus, human lifespan is not memoryless.

Common distributions with Memorylessness

Only Two standard distributions have this property.

Visualization

When we have new knowledge "given X lasted $s$ time units", the PDF is no longer same. Mathematically, the new conditional formula for $x \ge s$ is:

$$f(x \mid X > s) = \frac{f(x)}{P(X > s)}$$

https://www.desmos.com/calculator/xaxksndvgx