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Transcript
ECONOMETRICS I
APPENDIX A: A REVIEW OF SOME
STATISTICAL CONCEPTS
Textbook: Damodar N. Gujarati (2004) Basic Econometrics,
4th edition, The McGraw-Hill Companies
A.1: SUMMATION AND PRODUCT
OPERATORS
A.1: SUMMATION AND PRODUCT
OPERATORS
A.2 SAMPLE SPACE, SAMPLE POINTS
AND EVENTS
• The set of all possible outcomes of a random,
or chance, experiment is called the
population, or sample space, and each
member of this sample space is called a
sample point.
• Example: Remember tossing a coin… head, tail… the sample
space consists of these four possible outcomes: HH, HT, TH,
and TT.
• An event is a subset of the sample space.
A.2 SAMPLE SPACE, SAMPLE POINTS
AND EVENTS
• An event is a subset of the sample space.
• Events are said to be mutually exclusive if the
occurrence of one event precludes the
occurrence of another event.
• Events are said to be (collectively) exhaustible
if they exhaust all the possible outcomes of an
experiment.
• Thus, in the example, the events (a) two heads, (b) two tails, and (c)
one tail, one head exhaust all the outcomes; hence they are
(collectively) exhaustive events.
A.3 PROBABILITY AND
RANDOM VARIABLES
• Let A be an event in a sample space. By P(A),
the probability of the event A, we mean the
proportion of times the event A will occur in
repeated trials of an experiment.
A.3 PROBABILITY AND
RANDOM VARIABLES
• A variable whose value is determined by the
outcome of a chance experiment is called a
random variable (rv). Random variables are
usually denoted by the capital letters X, Y, Z,
and so on, and the values taken by them are
denoted by small letters x, y, z, and so on. A
random variable may be either discrete or
continuous.
A.3 PROBABILITY AND
RANDOM VARIABLES
• A discrete rv takes on only a finite (or
countably infinite) number of values.
For example, in throwing two dice, each
numbered 1 to 6, if we define the random
variable X as the sum of the numbers showing
on the dice, then X will take one of these
values: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, or 12.
Hence it is a discrete random variable.
A.3 PROBABILITY AND
RANDOM VARIABLES
• A continuous rv, on the other hand, is one
that can take on any value in some interval of
values..
The height of an individual is a continuous
variable—in the range, say, 60 to 65 inches it
can take any value, depending on the
precision of measurement.
A.4 PROBABILITY DENSITY FUNCTION
(PDF)
• Probability Density Function of a Discrete
Random Variable
A.4 PROBABILITY DENSITY FUNCTION
(PDF)
• Probability Density Function of a Continuous
Random Variable
A.4 PROBABILITY DENSITY FUNCTION
(PDF)
• Probability Density Function of a Continuous
Random Variable (Figure A.2)
P(a ≤ x ≤ b) means the probability that X lies in the
interval a to b.
A.4 PROBABILITY DENSITY FUNCTION
(PDF)
• Joint Probability Density Functions
A.4 PROBABILITY DENSITY FUNCTION
(PDF)
• Joint Probability Density Functions
A.5 CHARACTERISTICS OF PROBABILITY
DISTRIBUTIONS – Expected Value
A.5 CHARACTERISTICS OF PROBABILITY
DISTRIBUTIONS – Variance
A.5 CHARACTERISTICS OF PROBABILITY
DISTRIBUTIONS – Covariance
A.6 SOME IMPORTANT THEORETICAL
PROBABILITY DISTRIBUTIONS
•
•
•
•
Normal distribution (Z)
The χ2 (Chi-Square) Distribution
Student’s t Distribution
The F Distribution