Example
Suppose X and Y are two independent tosses of a fair coin, where we designate 1 for heads and 0 for tails. Let the third random variable Z be equal to 1 if one and only one of those coin tosses resulted in "heads", and 0 otherwise. Then jointly the triple (X, Y, Z) has the following probability distribution:
It is easy to verify that
- X and Y are independent, and
- X and Z are independent, and
- Y and Z are independent, however
- jointly X, Y, and Z are not independent, since any of them is completely determined by the other two (any of X, Y, Z is the sum (modulo 2) of the others). That is as far from independence as random variables can get. However, X, Y, and Z are pairwise independent, i.e. in each of the pairs (X, Y), (X, Z), and (Y, Z), the two random variables are independent.
Read more about this topic: Pairwise Independence
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“Our intellect is not the most subtle, the most powerful, the most appropriate, instrument for revealing the truth. It is life that, little by little, example by example, permits us to see that what is most important to our heart, or to our mind, is learned not by reasoning but through other agencies. Then it is that the intellect, observing their superiority, abdicates its control to them upon reasoned grounds and agrees to become their collaborator and lackey.”
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