UCLA; Economics 143; Cameron; Handout #2

UNIVERSITY OF CALIFORNIA, LOS ANGELES Department of Economics
Economics 143 (Cameron) - Applied Regression Analysis
Classroom Handout #2: Properties of Distributions

A. Expected Value

Intuitively, E[X] can be considered as a probability-weighted average of the values of the random variable; the center of gravity of a distribution. It is also called m_X, or the population mean.

E[X] = S _X x f(x) if X discrete E[X] = ò _X x f(x) dx if X continuous

Its properties are:

1. E[b] = b
2. E[X+Y] = E[X] + E[Y]
3. E[aX] = a E[X]
4. E[aX+b] = a E[X] + b
5. E[aX+bY] = a E[X] + b E[Y]

Unfortunately, you must also remember the following:

1. E[X/Y] is not equal to E[X]/E[Y]
2. E[XY] is not equal to E[X] E[Y] unless X and Y are independent
3. E[g(X)] is not equal to g(E[X]) except when g(X) is linear

B. Variance (and Standard Deviation)

Variance is a measure of the amount of dispersion or "spread" in a distribution. It is measured in units that are the square of the units in which the variable is measured, so standard deviation (the square root of variance) is often more intuitive. Standard deviation is measured in the same units as the variable itself.

Var[X] = s² = E[X-E[X]]² = E[X-m _X]² = E[X²] - m² St.Dev[X] = s _X = positive square root of Var[X] Properties include:

1. Var[a] = 0 (if "a" is a constant)
2. If X and Y are independent:
            Var[X+Y] = Var[X] + Var[Y]
            Var[X-Y] = Var[X] + Var[Y] (still a sum)
3. Var[X+b] = Var[X] (constant b has no variance)
4. Var[aX] = a² Var[X]
5. Var[aX+b] = a² Var[X]
6. If X and Y are independent:
            Var[aX+bY] = a² Var[X] + b² Var[Y]
7. If X and Y are not independent:
        Var[aX+bY] = a² Var[X] + b² Var[Y] + 2ab Cov[X,Y]
        But what is Cov[X,Y]?

C. Covariance

Measures the extent to which two variables move together. If E[X] = m _X and E[Y] = m _Y, then Cov[X,Y] = E[(X-m _X)(Y-m _Y)] = E[XY] - E[X]E[Y]. For discrete random variables, the formula is given by:

Cov[X,Y] = { S _X S _Y xy f(x,y) } - m _X m _Y Properties of covariance include:

1. If X and Y are independent, their covariance is zero.
2. Cov[ a+bX, c+dY ] = bd Cov[X,Y]
3. Cov[X,X] = Var[X]

D. Correlation

Covariances have associated units that are the product of the units of X and the units of Y. If X is in feet and Y is in pounds, Cov[X,Y] is measured in foot-pounds. If is often desirable to have a unit-free measure of the extent of the linear relationship between any two variables; use correlation.

r _X,Y = Cov[X,Y] / (s _X s _Y) Properties:

1. sign(r _X,Y) = sign(Cov[X,Y])
2. measures only the degree of LINEAR association, not slope
3. -1 < r < +1
4. r_X,X = 1

Now, for statistically dependent random variables, we must allow for Cov[X,Y] nonzero, so the general and special-case variance formulas are:

            Var[aX+bY] = a² Var[X] + b² Var[Y] + 2ab Cov[X,Y] <=== memorize this
            Var[X+Y] = Var[X] + Var[Y] + 2 Cov[X,Y]
            Var[X-Y] = Var[X] + Var[Y] - 2 Cov[X,Y]

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Updated: January 20, 1998
Prepared by: Trudy Ann Cameron