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Independence is a fundamental notion in probability theory, as in statistics and the theory of stochastic processes.Two events are independent, statistically independent, or stochastically independent [1] if, informally speaking, the occurrence of one does not affect the probability of occurrence of the other or, equivalently, does not affect the odds.
Independent: Each observation will not affect the next one, which means the 52 results are independent from each other. In contrast, if each card that is drawn is kept out of the deck, subsequent draws would be affected by it (drawing one king would make drawing a second king less likely), and the observations would not be independent.
Each of two urns contains twice as many red balls as blue balls, and no others, and one ball is randomly selected from each urn, with the two draws independent of each other. Let A {\displaystyle A} and B {\displaystyle B} be discrete random variables associated with the outcomes of the draw from the first urn and second urn respectively.
Pairwise independence does not imply mutual independence, as shown by the following example attributed to S. Bernstein. [3]Suppose X and Y are two independent tosses of a fair coin, where we designate 1 for heads and 0 for tails.
In the following diagram, the two objections weaken the contention, while the reasons support the premise of the objection: A sample argument using objections. Some argument mapping conventions allow for perspicuous representation of inferences. [12] In the following diagram, box 2.1 represents an inference, labeled with the inference rule ...
The probability distribution of the sum of two or more independent random variables is the convolution of their individual distributions. The term is motivated by the fact that the probability mass function or probability density function of a sum of independent random variables is the convolution of their corresponding probability mass functions or probability density functions respectively.
Graphs of probability P of not observing independent events each of probability p after n Bernoulli trials vs np for various p.Three examples are shown: Blue curve: Throwing a 6-sided die 6 times gives a 33.5% chance that 6 (or any other given number) never turns up; it can be observed that as n increases, the probability of a 1/n-chance event never appearing after n tries rapidly converges to ...
A misleading [1] Venn diagram showing additive, and subtractive relationships between various information measures associated with correlated variables X and Y. The area contained by both circles is the joint entropy H(X,Y).