The Random Variable
The
random variable
assigns each event in a sample space to a point or a union of
intervals on the real line.
In this way we can speak of probability distributions as
functions of real variables rather than as functions on sets.
The notion is entirely formal,
and is intended to be natural, in that when a properly defined sample space involves
Euclidean space, that use is carried through in defining the random variable.
Thus we
have some function X: S
∇
such that X(s) = x for some s
∈
S and x
∈
∇
, when s is an
event that can be discretely defined.
In defining the random variable, we also must
observe the notion of
equivalence
in the sense that if P[{s}] = b, then P[{X(s)}] = b also.
Example:
(Problem 2, p 174)
Sample space S contains 5 letters: S = {a, b, c, d, e}.
These letters are encoded via binary
strings: a = 1, b = 01, c = 001, d = 0001 and e = 0000.
Probabilities have been assigned
to the sample space as follows: p(a) = ½, p(b) = ¼, p(c) = 1/8, and p(d) = p(e) = 1/16.
We define a random variable Y to be equal to the length of the binary string.
Then
Y(a) = 1, Y(b) = 2, Y(c) = 3, and Y(d) = Y(e) = 4.
We see that S
Y
= {1, 2, 3, 4}.
Furthermore, we have the probability assignments:
p(1) = ½, p(2) = ¼, p(3) = 1/8 and p(4) = 1/8.
Example:
(Problem 4, p 174 – 5.)
The sample space is a circle of radius 1.
The random variable assigns to each point in the
circle its distance from the origin.
Then S
Y
= {y
∈
∇
0
≤
y < 1}.
Consider the event A
Y
⊆
S
Y
where A
Y
= {y
∈
∇
y < y
0
}.
We want to find the equivalent subset A
⊆
S.
First, we
note that S = {(a, b)
∈
∇
2
a
2
+ b
2
< 1}.
Then the requirement that Y(A) = A
Y
means that
A = {(a, b)
∈
∇
2
a
2
+ b
2
< y
0
}.
Finally, we can give meaning to probabilities applied to
either S or S
Y
if we make an assumption such as the random selection of points in the unit
circle.
Under such an assumption, we can relate the probability of an event space to the
area of the region covered by the space compared to the area of the entire sample space.
Since the area of the unit circle is
π
, the probability assigned to event space A
Y
is P[A
Y
] =
π
y
0
2
/
π =
y
0
2
.