Probability Of Error
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removed. (December 2009) (Learn how and when to remove this template message) In statistics, the term "error" arises in two ways. Firstly, it arises in the context of decision making, where the probability probability of error in digital communication of error may be considered as being the probability of making a wrong probability of error formula decision and which would have a different value for each type of error. Secondly, it arises in the context of statistical probability of error and bit error rate modelling (for example regression) where the model's predicted value may be in error regarding the observed outcome and where the term probability of error may refer to the probabilities of various amounts of
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error occurring. Hypothesis testing[edit] In hypothesis testing in statistics, two types of error are distinguished. Type I errors which consist of rejecting a null hypothesis that is true; this amounts to a false positive result. Type II errors which consist of failing to reject a null hypothesis that is false; this amounts to a false negative result. The probability of error is similarly distinguished. For a Type I probability of errors in measurement error, it is shown as α (alpha) and is known as the size of the test and is 1 minus the specificity of the test. It should also be noted that α (alpha) is sometimes referred to as the confidence of the test, or the level of significance (LOS) of the test. For a Type II error, it is shown as β (beta) and is 1 minus the power or 1 minus the sensitivity of the test. Statistical and econometric modelling[edit] The fitting of many models in statistics and econometrics usually seeks to minimise the difference between observed and predicted or theoretical values. This difference is known as an error, though when observed it would be better described as a residual. The error is taken to be a random variable and as such has a probability distribution. Thus distribution can be used to calculate the probabilities of errors with values within any given range. This statistics-related article is a stub. You can help Wikipedia by expanding it. v t e Retrieved from "https://en.wikipedia.org/w/index.php?title=Probability_of_error&oldid=721278136" Categories: ErrorStatistical modelsStatistics stubsHidden categories: Articles lacking sources from December 2009All articles lacking sourcesAll stub articles Navigation menu Personal tools Not logged inTalkContributionsCreate accountLog in Namespaces Article Talk
measure Complementary event Joint probability Marginal probability Conditional probability Independence Conditional independence Law of total probability Law of large numbers Bayes' theorem Boole's inequality Venn diagram Tree diagram v t e Pairwise error probability is the probability of error statistics error probability that for a transmitted signal ( X {\displaystyle X} ) its corresponding
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but distorted version ( X ^ {\displaystyle {\widehat {X}}} ) will be received. This type of probability is called ″pair-wise
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error probability″ because the probability exists with a pair of signal vectors in a signal constellation.[1] It's mainly used in communication systems.[1] Contents 1 Expansion of the definition 2 Closed form computation 3 https://en.wikipedia.org/wiki/Probability_of_error See also 4 References 5 Further reading Expansion of the definition[edit] In general, the received signal is a distorted version of the transmitted signal. Thus, we introduce the symbol error probability, which is the probability P ( e ) {\displaystyle P(e)} that the demodulator will make a wrong estimation ( X ^ ) {\displaystyle ({\widehat {X}})} of the transmitted symbol ( X ) {\displaystyle (X)} based on the https://en.wikipedia.org/wiki/Pairwise_error_probability received symbol, which is defined as follows: P ( e ) ≜ 1 M ∑ x P ( X ≠ X ^ | X ) {\displaystyle P(e)\triangleq {\frac {1}{M}}\sum _{x}\mathbb {P} (X\neq {\widehat {X}}|X)} where M is the size of signal constellation. The pairwise error probability P ( X → X ^ ) {\displaystyle P(X\to {\widehat {X}})} is defined as the probability that, when X {\displaystyle X} is transmitted, X ^ {\displaystyle {\widehat {X}}} is received. P ( e | X ) {\displaystyle P(e|X)} can be expressed as the probability that at least one X ^ ≠ X {\displaystyle {\widehat {X}}\neq X} is closer than X {\displaystyle X} to Y {\displaystyle Y} . Using the upper bound to the probability of a union of events, it can be written: P ( e | X ) ≤ ∑ X ^ ≠ X P ( X → X ^ ) {\displaystyle P(e|X)\leq \sum _{{\widehat {X}}\neq X}P(X\to {\widehat {X}})} Finally: P ( e ) = 1 M ∑ X ∈ S P ( e | X ) ≤ 1 M ∑ X ∈ S ∑ X ^ ≠ X P ( X → X ^ ) {\displaystyle P(e)={\tfrac {1}{M}}\sum _{X\in S}P(e|X)\leq {\tfrac {1}{M}
FeaturesTrial versionPurchaseCustomers Companies UniversitiesTraining and Consulting Course ListingCompanyArticlesHome > Articles > Calculating Type I Probability Calculating Type I Probability by Philip MayfieldI have had many requests to explain the math behind the statistics in the article Roger Clemens and a Hypothesis Test. The math is usually handled by http://www.sigmazone.com/Clemens_HypothesisTestMath.htm software packages, but in the interest of completeness I will explain the calculation in more detail. A t-Test provides the probability of making a Type I error (getting it wrong). If you are familiar with Hypothesis testing, then you can skip the next section and go straight to t-Test hypothesis. Hypothesis TestingTo perform a hypothesis test, we start with two mutually exclusive hypotheses. Here’s an example: when probability of someone is accused of a crime, we put them on trial to determine their innocence or guilt. In this classic case, the two possibilities are the defendant is not guilty (innocent of the crime) or the defendant is guilty. This is classically written as…H0: Defendant is ← Null HypothesisH1: Defendant is Guilty ← Alternate HypothesisUnfortunately, our justice systems are not perfect. At times, we let the guilty go probability of error free and put the innocent in jail. The conclusion drawn can be different from the truth, and in these cases we have made an error. The table below has all four possibilities. Note that the columns represent the “True State of Nature” and reflect if the person is truly innocent or guilty. The rows represent the conclusion drawn by the judge or jury.Two of the four possible outcomes are correct. If the truth is they are innocent and the conclusion drawn is innocent, then no error has been made. If the truth is they are guilty and we conclude they are guilty, again no error. However, the other two possibilities result in an error.A Type I (read “Type one”) error is when the person is truly innocent but the jury finds them guilty. A Type II (read “Type two”) error is when a person is truly guilty but the jury finds him/her innocent. Many people find the distinction between the types of errors as unnecessary at first; perhaps we should just label them both as errors and get on with it. However, the distinction between the two types is extremely important. When we commit a Type I error, we put an
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