Memorylessness

 ( Theory Of Probability Distributions ) 

In probability and statistics, memorylessness is a property of probability distributions. It describes situations where previous failures or elapsed time does not affect future trials or further wait time. Only the geometric and exponential distributions are memoryless.

---

Definition A random variable $X$ is memoryless if $$\backslash Pr(X>t+s\; \backslash mid\; X>s)=\backslash Pr(X>t)$$where $\backslash Pr$ is its probability mass function or probability density function when $X$ is discrete or continuous respectively and $t$ and $s$ are nonnegative numbers.

Frederik Michel Dekking (2025). 9781852338961, Springer London. . ISBN 9781852338961

Jim Pitman (1993). 9780387945941, Springer New York. . ISBN 9780387945941

In discrete cases, the definition describes the first success in an infinite sequence of independent and identically distributed Bernoulli trial, like the number of coin flips until landing heads.

Werner Nagel (2017). 9781119243526, Wiley. . ISBN 9781119243526

In continuous situations, memorylessness models random phenomena, like the time between two earthquakes.

Esra Bas (2025). 9783030323226, Springer International Publishing. . ISBN 9783030323226

The memorylessness property asserts that the number of previously failed trials or the elapsed time is independent, or has no effect, on the future trials or lead time.

The equality characterizes the geometric and exponential distributions in discrete and continuous contexts respectively.

In other words, the geometric random variable is the only discrete memoryless distribution and the exponential random variable is the only continuous memoryless distribution.

In discrete contexts, the definition is altered to $\backslash Pr(X>t+s\; \backslash mid\; X\; \backslash geq\; s)=\backslash Pr(X>t)$ when the geometric distribution starts at $0$ instead of $1$ so the equality is still satisfied.

---

Characterization of exponential distribution If a continuous probability distribution is memoryless, then it must be the exponential distribution.

From the memorylessness property,$$\backslash Pr(X>t+s\; \backslash mid\; X>s)=\backslash Pr(X>t).$$The definition of conditional probability reveals that$$\backslash frac\{\backslash Pr(X\; >\; t\; +\; s)\}\{\backslash Pr(X\; >\; s)\}\; =\; \backslash Pr(X\; >\; t).$$Rearranging the equality with the survival function, $S(t)\; =\; \backslash Pr(X\; >\; t)$, gives$$S(t\; +\; s)\; =\; S(t)\; S(s).$$This implies that for any natural number $k$$$S(kt)\; =\; S(t)^k.$$Similarly, by dividing the input of the survival function and taking the $k$-th root,$$S\backslash left(\backslash frac\{t\}\{k\}\backslash right)\; =\; S(t)^\{\backslash frac\{1\}\{k\}\}.$$In general, the equality is true for any rational number in place of $k$. Since the survival function is continuous and rational numbers are Dense set in the Real number (in other words, there is always a rational number arbitrarily close to any real number), the equality also holds for the reals. As a result,$$S(t)\; =\; S(1)^t\; =\; e^\{t\; \backslash ln\; S(1)\}\; =\; e^\{-\backslash lambda\; t\}$$where $\backslash lambda\; =\; -\backslash ln\; S(1)\; \backslash geq\; 0$. This is the survival function of the exponential distribution.

---

Characterization of geometric distribution If a discrete probability distribution is memoryless, then it must be the geometric distribution.

From the memorylessness property,$$\backslash Pr(X>t+s\; \backslash mid\; X\backslash geq\; s)=\backslash Pr(X>t)$$The definition of conditional probability reveals that$$\backslash frac\{\backslash Pr(X\; >\; t\; +\; s)\}\{\backslash Pr(X\; \backslash geq\; s)\}\; =\; \backslash Pr(X\; >\; t)$$From this it can be proven by induction that $$\backslash Pr(X\; >\; kt)\; =\; \backslash Pr(X\; >\; 1)^k$$Then it follows that$$f\_X(x)=Pr(X\backslash leq\; x)=1-Pr(X>x)=1-Pr(X>1)^x$$ and if we let $$Pr(X>1)=1-p$$for some $0\backslash leq\; p\; \backslash leq\; 1$. we can easily see that X is geometrically distributed with some parameter p. in other words $$X\backslash sim\; Geo(p)$$

Post Comment