Page 31 - Computational Statistics Handbook with MATLAB

Chapter 2: Probability Concepts 17

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yy
Axiomsof
Axioms
Axioms of P Pr PPrr obaoba bbii ilit litlit lity y
Axiomsofof
Probabilities follow certain axioms that can be useful in computational statis-
tics. We let S represent the sample space of an experiment and E represent
some event that is a subset of S.
AXIOM 1
The probability of event E must be between 0 and 1:
0 ≤ PE() ≤ . 1

AXIOM 2

PS() = . 1

AXIOM 3
, , , ,
For mutually exclusive events, E 1 E 2 … E k
k
PE ∪( 1 E ∪ … ∪ E ) = ∑ PE() .
i
k
2
i = 1
Axiom 1 has been discussed before and simply states that a probability
must be between 0 and 1. Axiom 2 says that an outcome from our experiment
must occur, and the probability that the outcome is in the sample space is 1.
Axiom 3 enables us to calculate the probability that at least one of the mutu-
, , ,
ally exclusive events E 1 E 2 … E k occurs by summing the individual proba-
bilities.

2.3 Conditional Probability and Independence

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litylity
P
r
Conditional Conditional ConditionalConditional PP roob abbi ilitylity
P
a
b
rr obob
Conditional probability is an important concept. It is used to define indepen-
dent events and enables us to revise our degree of belief given that another
event has occurred. Conditional probability arises in situations where we
need to calculate a probability based on some partial information concerning
the experiment.
The conditional probability of event E given event F is defined as follows:
© 2002 by Chapman & Hall/CRC