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Thanks!

Posted By: Timothy Chow
Date: Monday, 11 October 2010, at 5:33 p.m.

In Response To: Thanks! (Casper van der Tak)

When you integrate f(x) dx to get a probability, you're not actually summing up the values of f(x). If you were, then when you integrated something like the normal distribution, you'd get something infinite, because you'd be summing up infinitely many nonzero values of f(x). Rather, you can informally think of the integral as summing up the values of f(x) dx, where f(x) dx is infinitesimal. f(x) is not the probability of x; it is a probability density function, and you can use it to estimate the probability of a value between x and x+h as being about f(x)*h when h is small.

Regarding Bob Koca's comment, it is true that you can't in general sum up infinitely many probabilities to get the probability of the union. I don't know if you got as far in your mathematical education as the distinction between "countable" and "uncountable" infinities; if you have, then it's permissible to sum up countably many probabilities, but not uncountably many probabilities. But now we're getting technical and I don't want to say more unless you want more details here.

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