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I have long thought that the usual textbook discussion of independence
is misleading. In the first place, the most common situation where we
encounter independent random variables is with a cartesian product of
two indpendent sample spaces. Example: I toss a die and a coin. I have
reasonable assumptions about the distributions of events in either case
and I wish to discuss joint events. I have tried in vain to find natural
examples of independent random variables in a smple space not
constructed as a cartesian product.
I think that introducing the word "independent" as a descriptor of
sample spaces and then carrying it on to the events in the product space
is much less likely to generate the confusion due to the common informal
description "Independent events don't have anything to do with each
other" and "Mutually exclusive events can't happen together."
Comments?
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