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| | Greetings! I am Myrtle Shroyer. One of the very best issues in the world for me is to do aerobics and now I'm attempting to earn money with it. For a whilst I've been in South Dakota and my parents reside close by. She is a librarian but she's always wanted her personal business.<br><br>my webpage ... [http://lewat.in/healthymealsdelivered68340 http://lewat.in/] |
| In [[probability theory]] and [[statistics]], two real-valued [[random variable]]s, ''X'',''Y'', are said to be '''uncorrelated''' if their [[covariance]], E(''XY'') - E(''X'')E(''Y''), is zero. A set of two or more random variables is called uncorrelated if each pair of them are uncorrelated. If two variables are uncorrelated, there is no linear relationship between them.
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| Uncorrelated random variables have a [[Pearson correlation coefficient]] of zero, except in the trivial case when either variable has zero [[variance]] (is a constant). In this case the correlation is undefined.
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| In general, uncorrelatedness is not the same as [[orthogonality]], except in the special case where either ''X'' or ''Y'' has an expected value of 0. In this case, the [[covariance]] is the expectation of the product, and ''X'' and ''Y'' are uncorrelated [[if and only if]] E(''XY'') = 0.
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| If ''X'' and ''Y'' are [[statistical independence|independent]], then they are uncorrelated. However, not all uncorrelated variables are independent. For example, if ''X'' is a continuous random variable [[uniform distribution (continuous)|uniformly distributed]] on [−1, 1] and ''Y'' = ''X''<sup>2</sup>, then ''X'' and ''Y'' are uncorrelated even though ''X'' determines ''Y'' and a particular value of ''Y'' can be produced by only one or two values of ''X''.
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| ==Example Of Dependence Without Correlation ==
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| ;Uncorrelated random variables are not necessarily independent
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| * Let ''X'' be a random variable that takes the value 0 with probability 1/2, and takes the value 1 with probability 1/2.
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| * Let ''Z'' be a random variable that takes the value -1 with probability 1/2, and takes the value 1 with probability 1/2.
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| * Let ''U'' be a random variable constructed as ''U=XZ''.
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| The claim is that ''U'' and ''X'' have zero covariance (and thus are uncorrelated), but are not independent.
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| Proof:
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| First note:
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| * <math>E[U] = 0\times1/2 + 1\times1/4 + (-1)\times 1/4 = 0</math>
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| * <math>E[X] = 0\times1/2+1\times1/2 = 1/2</math>
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| Now, by definition <math>\mathrm{cov}(U,X) = E[(U-E[U])(X-E[X])] = E[ U (X-1/2)] = E[X^2Z - (1/2)XZ] = E[X^2Z] - (1/2)E[XZ]</math>
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| * <math>E[X^2Z] = 0\times1/2 + 1\times1/4 + (-1)\times 1/4 = 0</math>
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| * <math>E[XZ] = E[U] = 0</math>
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| Therefore <math>\mathrm{cov}(U,X)=0</math>
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| A necessary condition for showing that ''U'' and ''X'' are independent is showing that for any number ''a'' and ''b'', <math>Pr(U=a|X=b) = Pr(U=a)</math>. We prove that this is not true. PIck ''a=1'' and ''b=0''.
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| * <math>Pr(U=1|X=0) = Pr(XZ=1|X=0) = 0</math>
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| * <math>Pr(U=1) = Pr(XZ=1) = 1/4 </math>
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| Thus <math>Pr(U=1|X=0)\ne Pr(U=1)</math> so ''U'' and ''X'' are not independent.
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| Q.E.D.
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| ==When uncorrelatedness implies independence==
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| There are cases in which uncorrelatedness does imply independence. One of these cases is the one in which both random variables are two-valued (so each can be linearly transformed to have a [[binomial distribution]] with ''n''=1).<ref>[http://www.math.uah.edu/stat/expect/Covariance.html Virtual Laboratories in Probability and Statistics: Covariance and Correlation], item 17.</ref> Further, two jointly normally distributed random variables are independent if they are uncorrelated,<ref>{{cite book|chapter=Chapter 5.5 Conditional Expectation|pages=185–186|title=Introduction to Probability and Mathematical Statistics|year=1992|last1=Bain|first1=Lee|last2=Engelhardt|first2=Max|edition=2nd|isbn=0534929303}}</ref> although this does not hold for variables whose marginal distributions are normal and uncorrelated but whose joint distribution is not joint normal (see [[Normally distributed and uncorrelated does not imply independent]]).
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| ==See also==
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| *[[Correlation and dependence]]
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| *[[Binomial distribution#Covariance between two binomials|Binomial distribution: Covariance between two binomials]]
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| ==References==
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| {{reflist}}
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| ==Further reading==
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| *''Probability for Statisticians'', Galen R. Shorack, Springer (c2000) ISBN 0-387-98953-6
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| [[Category:Covariance and correlation]]
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| [[Category:Statistical terminology]]
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| [[de:Korrelation]]
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Greetings! I am Myrtle Shroyer. One of the very best issues in the world for me is to do aerobics and now I'm attempting to earn money with it. For a whilst I've been in South Dakota and my parents reside close by. She is a librarian but she's always wanted her personal business.
my webpage ... http://lewat.in/