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Unformatted text preview: ntly, the mean and variance, of a single random variable
convey important information about the distribution of the variable, and the moments are often
simpler to deal with than pmfs, pdfs, or CDFs. Use of moments is even more important when
considering more than one random variable at a time. That is because joint distributions are much
more complex than distributions for individual random variables.
Let X and Y be random variables with ﬁnite second moments. Three important related quantities are:
the correlation: E [XY ]
the covariance: Cov(X, Y ) = E [(X − E [X ])(Y − E [Y ])]
Cov(X, Y )
Cov(X, Y )
the correlation coeﬃcient: ρX,Y =
=
.
σX σY
Var(X )Var(Y )
Covariance generalizes variance, in the sense that Var(X ) = Cov(X, X ). Recall that there are
useful shortcuts for computing variance: Var(X ) = E [X (X − E [X ])] = E [X 2 ] − E [X ]2 . Similar
shortcuts exist for computing covariances:
Cov(X, Y ) = E [X (Y − E [Y ])] = E [(X − E [X ])Y ] = E [XY ] − E [X ]E [Y ].
In particular, if ei...
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 Spring '08
 Zahrn
 The Land

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