Thread: Lecture #2
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Old 01-26-2010, 11:50 AM
jhussmann jhussmann is offline
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Default Dirac delta, gamma function

Dirac delta "function" -

General information can be found at
http://mathworld.wolfram.com/DeltaFunction.html
http://en.wikipedia.org/wiki/Dirac_delta_function

I use quotation marks around "function" because the Dirac delta is not technically a function. Measure theoretic digression - when a random variable has a distribution that doesn't have any mass concentrated at any single points, the distribution is called absolutely continuous, and we can come up with a nicely behaved probability density function for the distribution. If the distribution does place non-zero mass at any point, such a point is called an atom of the distribution, and we cannot technically come up with a p.d.f. for the distribution. Dirac deltas are as close as we can get - objects that are not functions, but behave in the ways we would expect a "real" p.d.f. to when we integrate it. In the context of this course, Dirac deltas are essentially notational shorthand that allow us to pretend that a distribution admits a density when it technically does not.

Gamma function -

A good discussion of the gamma function can be found in chapter 8 of Rudin's "Principles of Mathematical Analysis". If anyone is interested, I could scan the relevant pages for you (or you could get the book, which is a classic).
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