Chapter 10: Covariance and Correlation

Report
CIS 2033 BASED ON
DEKKING ET AL. A MODERN INTRODUCTION TO PROBABILITY AND STATISTICS. 2007
INSTRUCTOR LONGIN JAN LATECKI
CHAPTER 10:
COVARIANCE AND CORRELATION
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As an example, take g(x, y) = xy for
discrete random variables X and Y with
the joint probability distribution given in the
table. The expectation of XY is computed
as follows:
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With the rule above we can compute the expectation of a random
variable X with a Bin(n,p)
which can be viewed as sum of Ber(p) distributions:
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Proof that E[X + Y] = E[X] + E[Y]:
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Var(X + Y) is generally not equal to Var(X) + Var(Y)
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If Cov(X,Y) > 0 , then X and Y are positively correlated.
If Cov(X,Y) < 0, then X and Y are negatively correlated.
If Cov(X,Y) =0, then X and Y are
uncorrelated.
Gustavo Orellana
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Now let X and Y be two independent random variables.
Then Cov(X, Y ) = E[XY ] − E[X]E[Y ] = 0.
Hence, then X and Y are uncorrelated.
We proved that if X and Y are two independent random variables,
then they are uncorrelated.
In general, E[XY] is NOT equal to E[X]E[Y].
INDEPENDENT VERSUS UNCORRELATED.
If two random variables X and Y are independent, then
X and Y are uncorrelated.
The converse is not true as we will see on the next slide.
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Then Cov(X, Y ) = E[XY ] − E[X]E[Y ] = 0 and X and Y are uncorrelated,
but they are dependent.
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The variance of a random variable with a Bin(n,p) distribution:
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The covariance changes under a change of units
The covariance Cov(X,Y) may not always be suitable to express the dependence
between X and Y. For this reason, there is a standardized version of the
covariance called the correlation coefficient of X and Y, which remains unaffected
by a change of units and, therefore, is dimensionless.
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Correlation coefficient is also called Pearson
correlation coefficient.
(from Wikipedia) Examples of scatter diagrams with different
values of correlation coefficient.
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(from Wikipedia) Several sets of (x, y) points, with the correlation coefficient
of x and y for each set. Note that the correlation reflects the non-linearity
and direction of a linear relationship (top row), but not the slope of that
relationship (middle), nor many aspects of nonlinear relationships (bottom).
N.B.: the figure in the center has a slope of 0 but in that case the correlation
coefficient is undefined because the variance of Y is zero.
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