Chapter 2
Continuous Probability
Densities
2.1
Simulation of Continuous Probabilities
In this section we shall show how we can use computer simulations for experiments
that have a whole continuum of possible outcomes.
Probabilities
Example 2.1
We begin by constructing a spinner, which consists of a circle of
unit
circumference
and a pointer as shown in Figure 2.1. We pick a point on the circle
and label it 0, and then label every other point on the circle with the distance, say
x
, from 0 to that point, measured counterclockwise. The experiment consists of
spinning the pointer and recording the label of the point at the tip of the pointer.
We let the random variable
X
denote the value of this outcome. The sample space
is clearly the interval [0
,
1). We would like to construct a probability model in which
each outcome is equally likely to occur.
If we proceed as we did in Chapter 1 for experiments with a ﬁnite number of
possible outcomes, then we must assign the probability 0 to each outcome, since
otherwise, the sum of the probabilities, over all of the possible outcomes, would
not equal 1. (In fact, summing an uncountable number of real numbers is a tricky
business; in particular, in order for such a sum to have any meaning, at most
countably many of the summands can be diﬀerent than 0.) However, if all of the
assigned probabilities are 0, then the sum is 0, not 1, as it should be.
In the next section, we will show how to construct a probability model in this
situation. At present, we will assume that such a model can be constructed. We
will also assume that in this model, if
E
is an arc of the circle, and
E
is of length
p
, then the model will assign the probability
p
to
E
. This means that if the pointer
is spun, the probability that it ends up pointing to a point in
E
equals
p
, which is
certainly a reasonable thing to expect.
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