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Difference between revisions of "Positive correlation"

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Assuming finite variances, two random variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073870/p0738701.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073870/p0738702.png" /> are positively correlated if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073870/p0738703.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073870/p0738704.png" /> denotes the [[Mathematical expectation|mathematical expectation]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073870/p0738705.png" />. In case the conditional mean of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073870/p0738706.png" /> given <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073870/p0738707.png" /> is linear in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073870/p0738708.png" />, positive correlation implies that this conditional mean is increasing in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073870/p0738709.png" />.
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Assuming finite [[variance]]s, two random variables $X$ and $Y$ are positively correlated if $\mathbf{E}(XY) > \mathbf{E}X\,\mathbf{E}Y$, where $\mathbf{E}X$ denotes the [[mathematical expectation]] of $X$. In case the conditional mean of $X$ given $Y$ is linear in $Y$, positive correlation implies that this conditional mean is increasing in $Y$.
  
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See [[Correlation (in statistics)|Correlation (in statistics)]].
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See [[Correlation (in statistics)]].
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Latest revision as of 17:19, 20 September 2017

Assuming finite variances, two random variables $X$ and $Y$ are positively correlated if $\mathbf{E}(XY) > \mathbf{E}X\,\mathbf{E}Y$, where $\mathbf{E}X$ denotes the mathematical expectation of $X$. In case the conditional mean of $X$ given $Y$ is linear in $Y$, positive correlation implies that this conditional mean is increasing in $Y$.

See Correlation (in statistics).

How to Cite This Entry:
Positive correlation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Positive_correlation&oldid=41903
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