Statistical Independence Relation

A Statistical Independence Relation is a Binary Relation that determines whether the Probability Distribution of one Random Variable is unrelated to the Probability Distribution of the other Random Variable.

• AKA: Independence Relation.
• Context:
  • It can be expressed as: P(A∩B) = P(A)P(B) or P(A|B)=P(A).
  • Two events A and B are statistically independent if the chance that they both happen simultaneously is the product of the chances that each occurs individually..
  • Alternatively, two events are independent if the discovery that one of the event has occurred does not help you determine whether the other event has also occurred: i.e., P(A|B) = P(A).
• Example(s):
  • In a Binomial Process, such as a String of Coin Toss Experiment, each Random Variable representing the Outcome of each Experiment is an a Statistical Independence Relation with each other.
• See: Conditional Probability, Independence Assumption, Mutual Independence Relation, Distinct Set Relation, Statistical Dependence Relation, IID Random Variable Set.

References

• (Wikipedia, 2009) http://en.wikipedia.org/wiki/Statistical_independence
  • In probability theory, to say that two events are independent intuitively means that the occurrence of one event makes it neither more nor less probable that the other occurs. For example:
    • The event of getting a 6 the first time a die is rolled and the event of getting a 6 the second time are independent.
    • By contrast, the event of getting a 6 the first time a die is rolled and the event that the sum of the numbers seen on the first and second trials is 8 are dependent.
    • If two cards are drawn with replacement from a deck of cards, the event of drawing a red card on the first trial and that of drawing a red card on the second trial are independent.
    • By contrast, if two cards are drawn without replacement from a deck of cards, the event of drawing a red card on the first trial and that of drawing a red card on the second trial are dependent.
  • Similarly, two random variables are independent if the conditional probability distribution of either given the observed value of the other is the same as if the other's value had not been observed. The concept of independence extends to dealing with collections of more than two events or random variables.
  • In some instances the term "independent" is replaced by "statistically independent", "marginally independent" or "absolutely independent"[1]

2006

• Suzanne R. Dubnicka. (2006). "STAT 510: Handout 1 - Probability Terminology. Kansas State University
  • TERMINOLOGY : We call two events A and B mutually exclusive, or disjoint, if A \ B = ; so that they have no outcomes in common. Thus, if A occurs then B cannot occur. Extending this definition to a finite or countable collection of sets is obvious.
• Suzanne R. Dubnicka. (2006). "STAT 510: Handout 2 - Counting Techniques and More Probabililty. Kansas State University
  • TERMINOLOGY : When the occurrence or non-occurrence of A has no effect on whether or not B occurs, and vice-versa, we say that the events A and B are independent. Mathematically, we define A and B to be independent iff (if and only if)
    • P(A ∩ B) = P(A)P(B).
  • Otherwise, A and B are called dependent events. Note that if A and B are independent,
    • P(A|B) = P(A ∩ B) / P(B) = P(A)P(B)/P(B) = P(A)
  • and
    • P(B|A) = P(B ∩ A) / P(A) = P(B)P(A) / P(A) = P(B).