Lecture #2

[wpress]

Attached are the slides for Lecture #2 as given on 1/25.

[jhussmann]

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).

[johnwoods]

I would love to see a copy of those pages -- on the gamma function. I've used them before, but they were never actually *explained*.

[jhussmann]

I will get them scanned ASAP. In the mean time, here is an article cited in the pages on the historical development of the gamma function that is a good read - http://www.jstor.org/stable/2309786.

[johnwoods]

I wish we had learned about the gamma function in terms of its history rather than its characteristics. I think the former would have helped me remember the latter.