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Introduction to Probability
One key assumption we make is that
is a
-algebra containing
, meaning that countably many complements, unions, and intersections of events in
are also events in
. The probability measure
must obey Kolmogorov’s Axioms.
- 1.
- 2.
- 3.Ifand, then
We choose
and
to model problems in a way that makes our calculations easy.
Intuitively, conditional probabilty is the probability of event
given that event
has occurred. In terms of probability spaces, it is as if we have taken
and now have a probabilty measure
belonging to the space
.
If
, then
are independent if and only if
. In other words, knowing
occurred gave no extra information about
.
Conditional independence is a special case of independence where
and
are not necessarily independent in the original probability space which has the measure
, but are independent in the new probability space conditioned on
with the measure
.
Last modified 2yr ago