INTRODUCTION TO ECONOMETRICS II ECO 306 NOUN 20
X P X P 2 3
4 5
6 7
8 9
10 11 12
( ) ∑ In the case of the two dice,
the values … were the numbers 2 … 12:
= 2,
=
3...
=12, and
=
,
=
=
. As shown in table 1.0, the expected value is
7. Also, the expected value of a random variable is described as population mean. In the case of the random variable
X, the
population mean is given as1.1.3.3 Expected value rules There are three main rules of expected values that are equally valid for both discrete and continuous random variables. These are
Rule 1: The expected value of the sum of several variables is equal to the sum of their respective expected values. For example, if you
have three random variables X,
Y, and
Z,
( ) ( ) ( ) ( )
…[1.02]
INTRODUCTION TO ECONOMETRICS II ECO 306 NOUN 21
Rule 2: If you multiply a random variable by a constant, you multiply its expected value by the same constant. If
X is
a random variable and b is a constant,
( ) ( )
…[1.03]
Rule 3: The expected value of a constant is that constant. For example, if
b is a constant.
( )
…[1.04] Putting the three rules together
suppose we wish to calculate E(
Y), where we have
…[1.05] and and are constants. Then,
( ) (
)
…[1.06]
(
) (
) ( )
…[1.07
=
( ) ( )…[1.08]
1.1.3.4 Sampling theory The goals of a sample survey and an experiment are very different. The role of randomisation also differs.
In both cases, without randomisation, there can be no inference. Without randomisation, the analyst can only describe the observations and cannot generalise the results.
In the sample survey, randomisation is used to reduce bias and to allow the results of the sample to be generalised to the population from which the sample was drawn. In an experiment, randomisation is used to balance the effects of confounding variables. The objective of asample survey is often to estimate a population mean and variance.
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