Memoryless Property Of Geometric Distribution

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The memoryless holding (also chosen the forgetfulness property) ways that a given probability distribution is independent of its history. Any fourth dimension 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 Holding

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 Ten is greater than a is simply the probability of X being greater than b. Symbolically, we write:

P( Ten > a + b | x > a ) = P ( 10 > b )

To make this specific, allow a be 5 and b 10. If our probability distribution is memoryless, the probability X > 15 if we know X > 5 is exactly the aforementioned every bit the probability of Ten existence greater than ten.

Note that this is not the same every bit the probability of Ten being greater than 15, as it would exist if the events X > xv and X > v were independent.

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

P( 10 > t + south | x > t ) = P ( ten > s ).

Examples of the Memoryless Property

Tossing a fair coin is an instance of probability distribution that is memoryless. Every time you toss the money, you lot take a 50 percent take chances of it coming up heads. It doesn't matter whether or non the last v times you threw the dice it came up consistently tails; the probability of heads in the next throw is ever going to be nix.

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 ofttimes or when your hardware failed in the past. The probability it will fail five minutes from now is independent of the fact that information technology hasn't failed for three months. The probability distribution tin exist 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 holding (over the set of reals) X is necessarily an exponential.

The detached geometric distribution (the distribution for which P(X = due north) = p(1 − p)^{n − 1}, for all northward≥1) is likewise memoryless.

Sources:

Conditional Probabilities and the Memoryless Belongings
Wolfram Memoryless

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