This module contains a set of definitions from statistics that might be useful for advanced undergraduates.
Important definitions in statistics
It is not unusual for students to forget important concepts learned in an earlier course. This set of definitions is intended to stir memories of those wonderful times when you were learning statistics and econometrics. It is not intended to replace a statistics course but to provide you with a handy guide to the denfinition of some important terms in the statistical tools used by economists.
Random variables
Random experiment
A random experiment is an experiment whose outcome is uncertain.
Outcome space
The outcome space (also sometimes referred to as the sample space) is the list of all possible outcomes of a random experiment.
Single toss of a coin.
Consider the toss of a coin. Since the outcome is uncertain, tossing the coin is an example of a random experiment. The outcome space consists of a heads and a tails. If we let
X be 0 if the outcome is a heads and let
X equal 1 if the outcome is a tails, then
X is a random variable. Since
X only can take on integer values (0 or 1), it is a discrete random variable.
Random variable
A random variable is a number that can be assigned to an outcome of a random experiment. A discrete random variable has a finite number of possible values while a continuous random variable has an infinite number of potential values.
Non-stochastic variable
A non-stochastic variable is any variable that is not a random variable; i.e., does not represent the outcome of a random experiment.
Multiple tosses of a coin.
Let
x equal the number of heads that occur when a coin is tossed n times. The tossing of the coin
n times is a random experiment. The outcome space of this random experiment is an integar between 0 and
n . Since the value
x is equal represents the outcome of a random experiment, it is a random variable.
Random sample
A random sample of size
n out of a population of size
N has the characteristic that every member of the population is equally likely to be chosen.
Height of college age women.
Consider a random sample of the population of college age women. The height,
x , of any woman chosen from this population is a random variable with a value somewhere in the outcome space, where the outcome space is a number between (say) 24 and 96 inches. Since in theory we can have as accurate a measurement as we might like,
x can be thought of as being a continuous random variable.
Probability
General terms
Probability distribution for a discrete random variable.
Consider a discrete random variable
that represents an outcome of the
n potential outcomes of a random experiment—that is, the set of potential outcomes is represented by
Any function is a probability if and only if (1)
(2)
for all i and j, and (3)
An example of a discrete distribution is in Example 4.
Discrete distribution.
Figure 1 illustrates a discrete probability distribution where
goes from 1 to 8. The areas in the shaded rectangles sum to 1.
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