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As an example, take g(x, y) = xy for discrete random variables X and Y withthe joint probability distribution given in the table. The expectation of XY is computed as follows:

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]:

Var(X + Y) is generally not equal to Var(X) + Var(Y)

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Gustavo Orellana 7

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.

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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.

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.