Independent Outcome Relation

A Independent Outcome Relation is Binary Relation between two Outcomes of the same Random Experiment that is True if the two Outcomes are independent.

• AKA: Independent Event, Independent Outcome, Independent Random Experiment Outcome.
  • …
• Example(s):
  • Two independent events A and B, the joint probability is [math]\displaystyle{ \mathrm{P}(A \cap B) = \mathrm{P}(A)\mathrm{P}(B) }[/math]
  • Two independent random variables X and Y, the joint cumulative distribution function is [math]\displaystyle{ F_{X,Y}(x,y) = F_X(x) F_Y(y), }[/math]
• Counter-Example(s):
  • Two dependent events, A and B, the joint probability is [math]\displaystyle{ \mathrm{P}(A \cap B) \neq \mathrm{P}(A)\mathrm{P}(B) }[/math]
  • Two dependent random variables X and Y, the joint cumulative distribution function is [math]\displaystyle{ F_{X,Y}(x,y) \neq F_X(x) F_Y(y), }[/math]
• See: Random Experiment Outcome, Conditional Probability Function, Conditional Independence Relation.

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References

2009a

• (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Independent_event | http://en.wikipedia.org/wiki/Independence_(probability_theory)
  • 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]

2009b

• http://www.mathchamber.com/algebra/docs/general/glossary.htm#I
  • Independent Event An occurrence or outcome that is not affected by previous occurrences or outcomes. The probability of tossing a coin heads or tails is an independent event. see dependent event
  • Independent Variable The domain contains values represented by the independent variable. The domain values are graphed on the x-axis (see dependent variable).

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