Info

Since E(X) = 4 (see Example 12), we finally have var (X) = 243/45 -

Properties of Variance

2. The variance of a constant is zero.

3. If a and b are constants, then var (aX + b) = a2 var (X)

4. If X and Y are independent random variables, then var (X + Y) = var (X) + var (Y) var (X - Y) = var (X) + var (Y)

This can be generalized to more than two independent variables.

5. If X and Y are independent rv's and a and b are constants, then var (aX + bY) = a2 var (X) + b2 var (Y)

Covariance

Let X and Y be two rv's with means ix and iy, respectively. Then the co-variance between the two variables is defined as cov (X, Y) = E{(X - ix)(Y - Xy)} = E(XY) - ixiy

It can be readily seen that the variance of a variable is the covariance of that variable with itself.

882 APPENDIX A: A REVIEW OF SOME STATISTICAL CONCEPTS

The covariance is computed as follows:

= XYf (*, y) - Px Py yx if X and Y are discrete random variables, and

if X and Y are continuous random variables.

Properties of Covariance

1. If X and Y are independent, their covariance is zero, for cov (X, Y) — E(XY) - /Xx fly

— fXfy - /xfy since E(XY) — E(X)E(Y) — // _ 0 when X and Y are independent

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