Databases Reference
In-Depth Information
of your getting killed while crossing a particular street is 0.1, you might wish to interpret this
information in a more subjective manner. The idea is to use the interpretation that gives you
the most insight into a particular problem, while remembering that your interpretation will not
change the mathematics of the situation.
Now that we have expended a lot of verbiage to say what probability is, let's spend a
few lines saying what it is not. Probability does not imply certainty. When we say that the
probability of an event is one, this does not mean that event
will
happen. On the other hand,
when we say that the probability of an event is zero, that does not mean that event
won't
happen.
Remember, mathematics only models reality, it is
not
reality.
A.2 Random Variables
When we are trying to mathematically describe an experiment and its outcomes, it is much
more convenient if the outcomes are numbers. A simple way to do this is to define a mapping or
function that assigns a number to each outcome. This mapping or function is called a
random
variable
. To put that more formally: Let
S
be a sample space with outcomes
{
ω
i
}
. Then the
random variable
X
is a mapping
X
:
S
→
R
(A.9)
where
R
denotes the real number line. Another way of saying the same thing is
X
(ω)
=
x
ω
∈
S
,
x
∈
R
(A.10)
The random variable is generally represented by an uppercase letter, and this is the con-
vention we will follow. The value that the random variable takes on is called the
realization
of the random variable and is represented by a lowercase letter.
ExampleA.2.1:
Let's take our television example and rewrite it in terms of a random variable
X
:
(
)
=
X
pr oductcommer ci al
0
X
(
ser
v
icecommercial
)
=
1
X
(
ne
w
s
)
=
2
X
(
pseudone
w
s
)
=
3
X
(
talksho
w)
=
4
X
(v
ariet y
)
=
5
X
(
comedy
)
=
6
X
(
ad
v
enture
)
=
7
X
(
mo
v
ie
)
=
8
Now, instead of talking about the probability of certain programs, we can talk about
the probability of the random variable
X
taking on certain values or ranges of values. For
