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“Principles of Econometrics”

When a distribution function F is di§erentiable we deÖne the probability density function
f(u) = d
duF(u):
The density contains the same information as the distribution function, but the density is typically
easier to visually interpret.
12
CHAPTER 2. CONDITIONAL EXPECTATION AND PROJECTION 13
Dollars per Hour
Wage Distribution
0 10 20 30 40 50 60 70
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Dollars per Hour
Wage Density
0 10 20 30 40 50 60 70 80 90 100
Figure 2.1: Wage Distribution and Density. All Full-time U.S. Workers
In Figure 2.1 we display estimates1 of the probability distribution function (on the left) and
density function (on the right) of U.S. wage rates in 2009. We see that the density is peaked around
$15, and most of the probability mass appears to lie between $10 and $40. These are ranges for
typical wage rates in the U.S. population.
Important measures of central tendency are the median and the mean. The median m of a
continuous2 distribution F is the unique solution to
F(m) = 1
2
:
The median U.S. wage ($19.23) is indicated in the left panel of Figure 2.1 by the arrow. The median
is a robust3 measure of central tendency, but it is tricky to use for many calculations as it is not a
linear operator.
The expectation or mean of a random variable y with density f is
 = E (y) = Z 1
1
uf(u)du:
Here we have used the common and convenient convention of using the single character y to denote
a random variable, rather than the more cumbersome label wage. A general deÖnition of the mean
is presented in Section 2.31. The mean U.S. wage ($23.90) is indicated in the right panel of Figure
2.1 by the arrow.
We sometimes use the notation Ey instead of E (y) when the variable whose expectation is being
taken is clear from the context. There is no distinction in meaning.
The mean is a convenient measure of central tendency because it is a linear operator and
arises naturally in many economic models. A disadvantage of the mean is that it is not robust4
especially in the presence of substantial skewness or thick tails, which are both features of the wage
distribution as can be seen easily in the right panel of Figure 2.1. Another way of viewing this
is that 64% of workers earn less that the mean wage of $23.90, suggesting that it is incorrect to
describe the mean as a ìtypicalîwage rate.