24 1. Preparatory material
1.1.4. Conditioning. Random variables are inherently non-deterministic
in nature, and as such one has to be careful when applying deterministic laws
of reasoning to such variables. For instance, consider the law of the excluded
middle: a statement P is either true or false, but not both. If this statement
is a random variable, rather than deterministic, then instead it is true with
some probability p and false with some complementary probability 1 p.
Also, applying set-theoretic constructions with random inputs can lead to
sets, spaces, and other structures which are themselves random variables,
which can be quite confusing and require a certain amount of technical care;
consider, for instance, the task of rigorously defining a Euclidean space
Rd
when the dimension d is itself a random variable.
Now, one can always eliminate these difficulties by explicitly working
with points ω in the underlying sample space Ω, and replacing every ran-
dom variable X by its evaluation X(ω) at that point; this removes all the
randomness from consideration, making everything deterministic (for fixed
ω). This approach is rigorous, but goes against the “probabilistic way of
thinking”, as one now needs to take some care in extending the sample
space.
However, if instead one only seeks to remove a partial amount of ran-
domness from consideration, then one can do this in a manner consistent
with the probabilistic way of thinking, by introducing the machinery of con-
ditioning. By conditioning an event to be true or false, or conditioning a
random variable to be fixed, one can turn that random event or variable into
a deterministic one, while preserving the random nature of other events and
variables (particularly those which are independent of the event or variable
being conditioned upon).
We begin by considering the simpler situation of conditioning on an
event.
Definition 1.1.18 (Conditioning on an event). Let E be an event (or state-
ment) which holds with positive probability P(E). By conditioning on the
event E, we mean the act of replacing the underlying sample space Ω with
the subset of Ω where E holds, and replacing the underlying probability
measure P by the conditional probability measure P(|E), defined by the
formula
(1.31) P(F |E) := P(F E)/P(E).
All events F on the original sample space can thus be viewed as events (F |E)
on the conditioned space, which we model set-theoretically as the set of all
ω in E obeying F . Note that this notation is compatible with (1.31).
All random variables X on the original sample space can also be viewed
as random variables X on the conditioned space, by restriction. We will
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