|WikiProject Statistics||(Rated B-class, Mid-importance)|
|WikiProject Mathematics||(Rated B-class, Mid-importance)|
How to prove that exponential distribution is the ONLY distribution that has the memoriless property?
- That is an excellent question, and at some point soon I'll add something to the article on this point. Here's the quick version of the answer:
- Let G(t) = Pr(X > t).
- Then basic laws of probability quickly imply that G(t) gets smaller as t gets bigger. The memorylessness of this distribution is expressed as
- Pr(X > t + s | X > t) = Pr(X > s).
- By the definition of conditional probability, this implies
- Pr(X > t + s)/Pr(X > t) = Pr(X > s).
- Thus we have the functional equation
- G(t + s) = G(t) G(s)
- AND we have the fact that G is a monotone decreasing function.
- The functional equation alone will imply that G restricted to rational multiples of any particular number is an exponential function. Combined with the fact that G is monotone, this implies G on its whole domain is an exponential function.
- That's a bit quick and hand-waving, but the detailed proof can be reconstructed from it. Michael Hardy 00:00, 13 November 2005 (UTC)
question: considering discrete-time processes, would the states be independent if the distribution of the values was Exponetial (or Geometric)? thanks, Akshayaj 19:56, 20 June 2006 (UTC)
- Regarding discrete and continuous stochastic processes, or indexed sets of random variables Xi, geometric and exponential distributions routinely model derived random variables rather than those Xi which constitute the process. For example, random waiting time, perhaps call it W, is the count of time points (for a discrete-time process) or the length of time interval (for a continuous-time process) before or between some occurrence(s) in the process Xi.
- Any sequence of geometric random variables Xi for all is an infinite discrete-time stochastic process. Any sequence of exponential random variables Xi for all is a finite discrete-time stochastic process. Those statements are true of any probability distributions, however, and they are silent regarding memorylessness or independence or stationarity.
- A sequence of exponential random variables Xi for all , to continue one example, may be called n observations of exponentially distributed data. If there is no further structure then it may not be fruitful to call the collection a discrete-time stochastic process in [0,infinity) --which is the range of an exponential r.v.-- but formally it is such a process.
- Last hour I rewrote the body of this article in terms of values m and n in a discrete index set and values t and s in a continuous index set, but retaining X in both cases for the random variable whose "memory" is under discussion. Those Xs are r.v. derived from the stochastic processes, if any. That is, they are functions of the Xs in this note. --P64 (talk) 21:08, 4 March 2010 (UTC)
In the lead section I have put a suggestive but shallow second paragraph in place of the following.
- wherein any derived probability from a set of random samples is distinct and has no information (i.e. "memory") of earlier samples.
What does that mean?
In order to improve this article much, we need commitment whether the subject pertains to probability or to something modeled by probability (perhaps a trial or a process or "sampling") or to both. This category error or category ambiguity plagues some neighboring articles, perhaps all of the articles with "process" in the title? --P64 (talk) 21:33, 4 March 2010 (UTC)
Memoryless has 2 different meanings that evolved separately in different branches of probability; one the Markov property , second the one related to conditional expectation when the distribution is applied to stopping time. We need to make sure both uses of the term are not mixed up Limit-theorem (talk) 10:50, 21 June 2013 (UTC)
- As far as I can tell, they are essentially the same - that the marginal probability is equal to the general probability of the outcome from current state. Just in a stochastic process, P(A) is meaningless, it has to be P(A|state). Markov memorylessness is simply that P(A|state&previous-states) is equal for all values of previous-states. SamBC(talk) 16:35, 17 June 2014 (UTC)