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