Chapter 2 Review
Statistics 519, Spring 1998

Expectations

Suppose X is a discrete random variable with values and density function p(x). The expectation of a function of X, expressed as g(X), is defined to be:

Note that the mean of X, , is E[X] (often denoted EX) and the variance of X, , is (often denoted V(X)).

As a convenience, we can show that

Linear Transformations

Since computing expectations of random variables involves either integration or summation, expectation is a linear operator, just like integration or summation. As a result, if A and B are constants,

If g(X)=X, then

From the preceding results, we can show

The Joint Behavior of X and Y

We want to extend the results on linear transformations of a single random variable X to linear transformations using a sample . To do this, we need to know something about the joint distribution of the sample. In actuality, we will only need results for two random variables, X and Y. We need to define joint density functions in order to define independence and covariance, which both arise when computing expectations of linear combinations of random variables. Assume X and Y are discrete. The joint density function of X and Y is p(x,y)=P(X=x,Y=y) for any (x,y). The marginal density functions of X and Y are, respectively,

and

The random variables X and Y are independent if

Covariance and Correlation

The covariance of X and Y is defined as and the correlation of X and Y is defined as

We can compute Cov(X,Y) using the shortcut formula

The random variables X and Y are uncorrelated if Cov(X,Y)=0 (or equivalently, . Independent random variables are uncorrelated, but uncorrelated variables are not necessarily independent. If X and Y are independent, then E[XY]=E[X]E[Y].

Expectations

If X and Y are independent

Covariances

If for , then

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