Probability review
February 3, 2011 Leave a comment
Basic probability review
- Sample space can be viewed as the set of results obtained from the probabilistic experiment under consideration. e.g. If the experiment consists of flipping a coin twice, then the sample space would consist of . Elements of are termed elementary events. Subsets of are termed events. e.g. is the event that the first coin flip results in a Head.
- Definition A field consists of a sample space and a set which is a collection of subsets of , and satisfies the following properties:
- closure under complement
- closure under union .
Note that closure under union and closure and complement imply closure under intersection (since ). Note also that . Elements of can be termed as observable events, and can be termed as an information structure.
- Definition Given a field , a probability measure is a function which satisfies:
- For all , .
- .
- For mutually disjoint events, , .
Some other properties that follow from the above:
- .
- .
- Definition A probability space consists of a field together with a probability measure defined on it.
- Principle of inclusion-exclusion
- Definition Condition probability of event given event is defined as :.
From this, we have that : .
- If denote a partition of the sample space , then, for any event , :
- Bayes’ rule If denote a partition of the sample space , then, for any event , :
- Definition A collection of events is \emph {independent} if for all subsets , :
- Equivalent defintion A collection of events is \emph {independent} if for all and for all subsets ,
- A collection of events is \emph {k-wise independent} if every set of events in the collection is independent. 2-wise independence is called pairwise independence.
- Definition A random variable is a real-valued function over the sample space such that for all :
- The distribution function for a random variable is defined as .
- The density function is defined as .
- The joint distribution function and joint density function are defined likewise :
.