Lecture Notes 1
Econ 410 – Introduction to Econometrics
1
Review of Probability
Random Variable
: a variable whose possible values are numerical outcomes of a random
phenomenon.
There are two types of random variables, discrete
and continuous
.
Ex.: # of students that will come to my office hours on Thursday (discrete); time at which the
lecture will end (continuous).
Outcomes
: the mutually exclusive values a random variable can take on.
Sample Space
: the set of all possible outcomes.
Event
: a subset of the sample space.
Ex.: rolling a dice
Sample Space = {1, 2, 3, 4, 5, 6} (all possible outcomes)
Event: “the number we get when rolling a dice is 4 or more” = {4, 5, 6}
Probability of an outcome
: the proportion of times the outcomes would occur in a very long
series of repetitions.
Ex.: When we roll a fair dice, each outcome has the same probability of occurring. This
probability is 1/6.
Remember that:
1) The probability of an outcome or event is always greater than or equal to 0.
2) The probability of the sample space is always equal to 1. In other words, the sum of the
probabilities of all the possible outcomes must be equal to 1.
Describing a discrete random variable
Probability distribution
: the list of all possible outcomes and the probability that each value
will occur.
Ex.: Probability distribution of the random variable Y = “# of students that will come to my
office hours next Tuesday” (for computational reasons, assume that 4 is the maximum
number of students I can meet. As soon as the 4
th
student leaves my office, I go home).
Y
Probability
Distribution
0
.50
1
.40
2
.06
3
.03
4
.01
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Lecture Notes 1
Econ 410 – Introduction to Econometrics
2
Probability that 2 students come to my office hours next Tuesday = Pr(
Y
=2) = .06
Probability of the event A: “3 or more students will come to my office hours next Tuesday” =
Pr(
A
) = Pr(
Y
=3) + Pr(
Y
=4) = .04
Probability of the event B: “an odd number of students will come to my office hours next
Tuesday” = Pr(
B
) = Pr(
Y
=1) + Pr(
Y
=3) = .43
Cumulative probability distribution
: the probability that the random variable is less than or
equal to a particular value.
90
.
)
1
Pr(
)
0
Pr(
)
1
Pr(
=
=
+
=
=
≤
Y
Y
Y
1
)
4
Pr(
=
≤
Y
Expected value (or mean)
: the long run average value of the random variable.
(
)
∑
=
=
=
k
i
i
i
Y
p
y
Y
E
1
µ
The expected value is just a weighted average of all the possible outcomes of the random
variable. The weight attached to each outcome is its probability.
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 Fall '08
 Staff
 Econometrics, Probability theory

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