Page 32 - Computational Statistics Handbook with MATLAB

18 Computational Statistics Handbook with MATLAB

CONDITIONAL PROBABILITY

(
(
PE F) = PE ∩ F) PF() > . 0 (2.5)
-----------------------;
PF()
Here PE ∩( F) represents the joint probability that both E and F occur
together and PF() is the probability that event F occurs. We can rearrange
Equation 2.5 to get the following rule:

MULTIPLICATION RULE

(
PE ∩ F) = PF()PE F) . (2.6)
(

eencence
Ind ependpend encence
eependpend
e
IndInd
e
Ind
Often we can assume that the occurrence of one event does not affect whether
or not some other event happens. For example, say a couple would like to
have two children, and their first child is a boy. The gender of their second
child does not depend on the gender of the first child. Thus, the fact that we
know they have a boy already does not change the probability that the sec-
ond child is a boy. Similarly, we can sometimes assume that the value we
observe for a random variable is not affected by the observed value of other
random variables.
These types of events and random variables are called independent. If
events are independent, then knowing that one event has occurred does not
change our degree of belief or the likelihood that the other event occurs. If
random variables are independent, then the observed value of one random
variable does not affect the observed value of another.
In general, the conditional probability PE F( ) is not equal to PE() . In these
cases, the events are called dependent. Sometimes we can assume indepen-
dence based on the situation or the experiment, which was the case with our
example above. However, to show independence mathematically, we must
use the following definition.

INDEPENDENT EVENTS
Two events E and F are said to be independent if and only if any of the following is
true:

PE ∩ F) = PE()PF(),
(
(2.7)
(
PE() = PE F).

© 2002 by Chapman & Hall/CRC