Lecture Notes
1
MSc in Operational Research
Introduction to Statistics
Lecture
9
Bivariate
Probability Distributions
Introduction
The intersection of two or more events is frequently of interest to an experimenter. For
example, the gambler playing blackjack is interested in the event of drawing both an ace and a
face card from a fiftytwocard deck. The biologist, observing the number of animals surviving
in a litter, is concerned with the intersection of these events:
A: The litter contains
n
animals
B: y animals survive.
Similarly, the observation of both height and weight on an individual represents the
intersection of a specific pair of heightweight measurements.
Most important to statisticians are the intersections that occur when sampling. Suppose that
Y
1
,Y
2
,
...,
Y
n
denote the outcomes on
n
successive trials of an experiment. For example, this
sequence could represent the weights of
n
people or the measurements of n physical
characteristics of a single person. A specific set of outcomes, or sample measurements, may
be expressed in terms of the intersection of the
n
events
),
y
Y
(
1
1
=
),
y
Y
(
2
2
=
….
),
y
Y
(
n
n
=
which we will denote as
).
y
......
,
y
,
y
,
y
(
n
3
2
1
Then to make inferences about the population from
which the sample was drawn, we will wish to calculate the probability of the intersection
).
y
......
,
y
,
y
,
y
(
n
3
2
1
A review of the role probability plays in making inferences, emphasizes the need for acquiring
the probability of the observed sample or, equivalently, the probability of the intersection of a
set of numerical events. Knowledge of this probability is fundamental to making an inference
about the population from which the sample was drawn. Indeed, this need motivates the
discussion of multivariate probability distributions.
Bivariate
Probability Distributions
Many random variables can be defined over the same sample space. For example, consider
the experiment of tossing a pair of dice. The sample space contains thirtysix sample points,
corresponding to the mn = (6)(6) = 36 ways which numbers may appear on the faces of the
dice. Any one of the following random variables could be defined over the sample space and
might be of interest to the experimenter:
Y,: The number of dots appearing on die 1.
Y
2
: The number of dots appearing on die 2.
Y
3
: The sum of the number of dots on the dice.
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Lecture Notes
2
Y
4
: The product of the number of dots appearing on the dice.
The thirtysix sample points associated with the experiment are equiprobable and
corresponding to the thirtysix numerical events
).
y
,
y
(
2
1
Thus throwing a pair of is would be
the simple event (1, 1). Throwing a 2 on die 1 and a 3 on die 2 would be the simple event (2,
3). Because all pairs
)
y
,
y
(
2
1
occur with the same relative frequency, we would assign a
probability of 1/36 to each sample point. For this simple example the intersection
)
y
,
y
(
2
1
contains only one sample point. Hence the bivariate probability function is
36
1
)
y
,
y
(
p
2
1
=
y
l
= 1, 2,..., 6; y
2
= 1, 2,..., 6
Definition
Let Y
1
and
Y
2
be discrete random variables. The
joint
(or bivariate)
probability
distribution
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 Winter '07
 Smith
 Statistics, Probability, Probability distribution, Probability theory, probability density function, Cumulative distribution function, yl

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