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| In [[probability theory]], if a [[random variable]] ''X'' has a [[binomial distribution]] with parameters ''n'' and ''p'', i.e., ''X'' is distributed as the number of "successes" in ''n'' independent [[Bernoulli trial]]s with probability ''p'' of success on each trial, then
| | They call me Vernita Bowker. My house at present in Georgia and by no means move. To cook issue he really enjoys . Hiring is how she creates a living but soon her husband and her start their own home office. Go to her website locate out more: http://phillylimorentals.com |
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| :<math>P(X\leq x) = P(X<x+1)</math>
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| for any ''x'' ∈ {0, 1, 2, ... ''n''}. If ''np'' and ''n''(1 − ''p'') are large (sometimes taken to mean ≥ 5), then the probability above is fairly well approximated by
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| :<math>P(Y\leq x+1/2)</math>
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| where ''Y'' is a [[normal distribution|normally distributed]] random variable with the same [[expected value]] and the same [[variance]] as ''X'', i.e., E(''Y'') = ''np'' and var(''Y'') = ''np''(1 − ''p''). This addition of 1/2 to ''x'' is a '''continuity correction'''.
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| A continuity correction can also be applied when other discrete distributions supported on the integers are approximated by the normal distribution. For example, if ''X'' has a [[Poisson distribution]] with expected value λ then the variance of ''X'' is also λ, and
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| :<math>P(X\leq x)=P(X<x+1)\approx P(Y\leq x+1/2)</math> | |
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| if ''Y'' is normally distributed with expectation and variance both λ.
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| ==Applications==
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| Before the ready availability of [[statistical software]] having the ability to evaluate probability distribution functions accurately, continuity corrections played an important role in the practical application of [[statistical hypothesis test|statistical tests]] in which the test statistic has a discrete distribution: it was a special importance for manual calculations. A particular example of this is the [[binomial test]], involving the [[binomial distribution]], as in [[checking whether a coin is fair]]. Where extreme accuracy is not necessary, computer calculations for some ranges of parameters may still rely on using continuity corrections to improve accuracy while retaining simplicity.
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| ==See also==
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| *[[Yates's correction for continuity]]
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| *[[Binomial_proportion_confidence_interval#Wilson_score_interval_with_continuity_correction|Wilson score interval with continuity correction]]
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| == References ==
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| * Devore, Jay L., ''Probability and Statistics for Engineering and the Sciences'', Fourth Edition, Duxbury Press, 1995.
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| * Feller, W., ''On the normal approximation to the binomial distribution'', The Annals of Mathematical Statistics, Vol. 16 No. 4, Page 319-329, 1945.
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| [[Category:Probability theory]]
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| [[Category:Statistical tests]]
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| [[Category:Computational statistics]]
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They call me Vernita Bowker. My house at present in Georgia and by no means move. To cook issue he really enjoys . Hiring is how she creates a living but soon her husband and her start their own home office. Go to her website locate out more: http://phillylimorentals.com