Chapter 5
Many Random Variables
•
A random variable models phenomena in which the experimental outcomes are
numbers
•
A random variable measures one physical parameter
•
Different random variables measure different parameters
•
Example:
Requests for different files arrive at a Web server.
Requests have two parameters of
interest: the time of arrival and the length of the data file requested
•
Sunspot activity causes different levels of fading of radio signals in different frequency bands
•
Different random variables measure different parameters
•
Experimental observations yield
vectors
or
sequences
or
arrays
of numbers
•
The observed vectors are random: we cannot predict beforehand which vector will be observed
next
•
The mathematical model of a random variable associates numbers with outcomes
ϖ
in a sample
space
•
Different random variables associate different numbers with each outcome
ϖ
•
[
X
(
ϖ
),
Y
(
ϖ
),
Z
(
ϖ
), … ] is a
vector
of observations
•
Example:
Ω
= set of people. One is chosen at random
X
= height of person chosen
Y
= weight
of person chosen
•
(
X
,
Y
) is a
random vector
or a
bivariate
random
variable
•
(
X
,
Y
) maps
ϖ
∈
Ω
onto the
pair
of numbers (
X
(
ϖ
),
Y
(
ϖ
))
•
(
X
,
Y
) maps
ϖ
∈
Ω
onto
point
(
X
(
ϖ
),
Y
(
ϖ
)) in the plane
•
Coordinate axes are u and v with value of
X
plotted along u axis and of
Y
along v axis
u
v
Ω
•
X
takes on three values,
Y
takes on only two values
•
Both
X
and
Y
individually are random variables
•
Together
,
X
and
Y
create a joint mapping of
ϖ
∈
Ω
onto the
random
point
(
X
(
ϖ
),
Y
(
ϖ
))
•
As usual, we drop the explicit dependence on
ϖ
when talking about random variables
•
(
X
,
Y
) is a random point in the plane
•
On succesive trials of the experiment, the outcomes are
ϖ
1
,
ϖ
2
,
ϖ
3,
…
and we observe the
random points (
X
(
ϖ
1
),
Y
(
ϖ
1
)), (
X
(
ϖ
2
),
Y
(
ϖ
2
)), (
X
(
ϖ
3
),
Y
(
ϖ
3
)), …
•
The u-coordinate of the random point is
X
(
ϖ
) and the v-coordinate is
Y
(
ϖ
)
© 1997 by Dilip V. Sarwate.
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