Probability 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 probability of error in digital communication making, where the probability of error may be considered as being the probability probability of error formula of making a wrong decision and which would have a different value for each type of error. Secondly, it probability of error and bit error rate arises in the context of statistical 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
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to the probabilities of various amounts of 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. probability of errors in measurement The probability of error is similarly distinguished. For a Type I 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: Arti
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 error probability probability of error statistics that for a transmitted signal ( X {\displaystyle X} ) its corresponding but
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distorted version ( X ^ {\displaystyle {\widehat {X}}} ) will be received. This type of probability is called ″pair-wise error probability″
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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 See also 4 https://en.wikipedia.org/wiki/Probability_of_error 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 received symbol, which is https://en.wikipedia.org/wiki/Pairwise_error_probability 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}}\sum _{X\in S}\sum _{{\widehat {X}}\neq X}P(X\to {\widehat {X}})} Closed for
Disorders Tests Fun & Games Select Page: Where to?Home Online Textbooks -- Psychology 101 -- Stats -- Research Methods -- Personality Synopsis Education Reference -- Timeline of Psychology -- Psychology Biographies -- Dictionary -- Books -- Guide to Online Psychology -- Psychotherapy Facts -- Psychotropic Medication Guide http://allpsych.com/researchmethods/errorprobability/ Disorders Tests Fun & Games advertisement AllPsychPsych Central's Virtual Psychology Classroom AllPsych > Research http://www.sigmazone.com/Clemens_HypothesisTestMath.htm Methods > Chapter 9.5 Probability of Error Chapter 9.5 Probability of Error By Dr. Christopher L. Heffner Dr. Christopher L. Heffner August 21, 2014 Chapter 9.5 Probability of Error2014-11-22T03:10:51+00:00 Probability of Error Since every score has some level of error researchers must decide how much error they are willing to accept prior to performing their research. This acceptable probability of error is then compared with the probability of error and if it is less, the study is said to be significant. For example, if we stated that we would accept 5% error at the onset of the study and our results indicated that the probability of error was 3%, we would reject the null hypothesis and state that the difference between the two groups was significant. If, however, the probability of error were shown probability of error to be 6%, we would accept the null hypothesis and state that the difference between the two groups was not significant. The probability of error is often abbreviated with a lower case ‘p,’ and the acceptable error is abbreviated with a lower case alpha (a). When we accept the null, then p > a, and when we reject the null, then p < = a. You will often see these symbols at the end of significance statements in research reports. While alpha can change, depending on the level set at the onset of the experiment, it should not change once the experiment begins. Common levels of acceptable error (referred to as significance) include, in order of use, 0.05, 0.01, 0.001, and 0.1. « Previous PageNext Page » Search Search for: advertisement AllPsych BlogTalking While Making Eye Contact Is Harder Than It LooksThe Mental Health Consequences of Natural DisastersBasketball Players Make More Shots When They Think About DeathThe Psychology of IronyHere’s What the Science Says About ClownsWho Lies the Most? 2016’s Ig Nobel Prize in PsychologyHigh School Background Might Have More to Do With College Success Than Intelligence or PersonalityMindfulness, Reappraisal, Emotion Suppression: Which Coping Strategies Work?Study Looks at Cat Body Language“Aha” Moments Can Be More Accurate Than AnalysisSectionsAbout AllPsych Books Crossword Puzzles Custom Feed Dictionary Disorders Education Fun &
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 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 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 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 innocent person in jail. When we commit a Type II error we let a guilty person go free. Which error is worse? The generally accepted position of society is that a Type I Error or putting an innocent person in