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outcome x = .5 and, because all outcomes are equally likely, P (X = x) = 0
for every x " [0, 1].
Because every x " [0, 1] is a possible outcome, our conclusion that P (X =
x) = 0 is initially somewhat startling. However, it is a mistake to identify
impossibility with zero probability. In Section 2.2, we established that the
impossible event (empty set) has probability zero, but we did not say that
it is the only such event. To avoid confusion, we now emphasize:
If an event is impossible, then it necessarily has probability zero;
however, having probability zero does not necessarily mean that
an event is impossible.
If P (X = x) = = 0, then the calculation in (4.2) reveals that the event
defined by (4.1) has probability zero. Furthermore, there is nothing special
about this particular event the probability of any countable event must be
zero! Hence, to obtain positive probabilities, e.g. P (X " [0, 1]) = 1, we must
consider events whose cardinality is more than countable.
Consider the events [0, .5] and [.5, 1]. Because all outcomes are equally
likely, these events must have the same probability, i.e.
P (X " [0, .5]) = P (X " [.5, 1]) .
Because [0, .5] *" [.5, 1] = [0, 1] and P (X = .5) = 0, we have
1 = P (X " [0, 1]) = P (X " [0, .5]) + P (X " [.5, 1]) - P (X = 0)
= P (X " [0, .5]) + P (X " [.5, 1]) .
Combining these equations, we deduce that each event has probability 1/2.
This is an intuitively pleasing conclusion: it says that, if outcomes are equally
4.1. A MOTIVATING EXAMPLE 83
likely, then the probability of each subinterval equals the proportion of the
entire interval occupied by the subinterval. In mathematical notation, our
conclusion can be expressed as follows:
Suppose that X(S) = [0, 1] and each x " [0, 1] is equally likely.
If 0 d" a d" b d" 1, then P (X " [a, b]) = b - a.
Notice that statements like P (X " [0, .5]) = .5 cannot be deduced from
knowledge that each P (X = x) = 0. To construct a probability distribution
for this situation, it is necessary to assign probabilities to intervals, not just
to individual points. This fact reveals the reason that, in Section 2.2, we
introduced the concept of an event and insisted that probabilities be assigned
to events rather than to outcomes.
The probability distribution that we have constructed is called the con-
tinuous uniform distribution on the interval [0, 1], denoted Uniform[0, 1]. If
X
" If y
F (y) = P (X d" y)
= P (X " (-", y])
= 0.
" If y " [0, 1], then
F (y) = P (X d" y)
= P (X " (-", 0)) + P (X " [0, y])
= 0 + (y - 0)
= y.
" If y > 1, then
F (y) = P (X d" y)
= P (X " (-", 0)) + P (X " [0, 1]) + P (X " (1, y))
= 0 + (1 - 0) + 0
= 1.
This function is plotted in Figure 4.1.
84 CHAPTER 4. CONTINUOUS RANDOM VARIABLES
-1 0 1 2
y
Figure 4.1: Cumulative Distribution Function of X
What about the pmf of X? In Section 3.1, we defined the pmf of a discrete
random variable by f(x) = P (X = x); we then used the pmf to calculate the
probabilities of arbitrary events. In the present situation, P (X = x) = 0 for
every x, so the pmf is not very useful. Instead of representing the probabilites
of individual points, we need to represent the probabilities of intervals.
Consider the function
ñø üø
ôø ôø
0 x " (-", 0)
òø ýø
f(x) = 1 x " [0, 1] , (4.3)
ôø ôø
óø þø
0 x " (1, ")
which is plotted in Figure 4.2. Notice that f is constant on X(S) = [0, 1], the
set of equally likely possible values, and vanishes elsewhere. If 0 d" a d" b d" 1,
then the area under the graph of f between a and b is the area of a rectangle
with sides b - a (horizontal direction) and 1 (vertical direction). Hence, the
area in question is
(b - a) · 1 = b - a = P (X " [a, b]),
1.0
0.5
F(y)
0.0
4.2. BASIC CONCEPTS 85
so that the probabilities of intervals can be determined from f. In the next
section, we will base our definition of continuous random variables on this
observation.
-1 0 1 2
x
Figure 4.2: Probability Density Function of X
4.2 Basic Concepts
Consider the graph of a function f : ’! , as depicted in Figure 4.3. Our
interest is in the area of the shaded region. This region is bounded by the [ Pobierz całość w formacie PDF ]