Introduction; Review of Random Variables
Probability :: Set Theory :: Definition :: Conditional :: Random :: Distribution :: R.V.S.
Conditional probability and independence
Conditional probability is perhaps the most important probability concept from an engineering point of view. It allows us to describe mathematically how our information changes when we are "given'' a measurement.
We define conditional probability as follows: Suppose
is a probability space and
with
. Define the conditional probability of
given
as
Essentially what we are saying is that the sample space is restricted from down to . Dividing by provides the correct normalization for this probability measure.
Some properties of conditional probability (consequences of the axioms of probability) are as follows:
 .

For
with
for
,
 .
The Law of Total Probability
: If
is a
partition of
and
, then
(Draw picture).
Bayes Formula is a simple formula for "turning around'' the conditioning. Because conditioning is so important in engineering, Bayes formula turns out to be a tremendously important tool (even though it is very simple). We will see applications of this throughout the semester.
Suppose
,
and
. Then
Why?
Is independent the same as disjoint?
Note: For , if and are independent then . (Since they are independent, can provide no information about , so the probability remains unchanged. If , then is independent of for any other event . (Why?).
The next idea is important in a lot of practical problem of
engineering interest.
(draw picture to illustrate the idea).