Correction on my last post

Both the title and body of my previous post referenced Class Activity 2, when I meant to refer to Class Activity 4 (CA4). Since you already turned in CA2, I hope this wasn't too confusing!

CA3 data

Part 6 of CA3 instructs you to construct a data table based on FV05. In the interest of time and uniformity, you should just use this table instead:

http://www.astro.caltech.edu/%7Ejohnjohn/astrostats/data/fv05_vf05_data.txt

The middle column indicates 0 if the star doesn't have a planet, and 1 if it does.

Parts 1-8 will be due Monday. Continue on to problems 9 if you feel particularly inspired to learn about how to deal with measurement uncertainties when doing this sort of analysis. 

Part 10 instructs you to write up your results. This is left over from last year. This year I've already encouraged you to make your results clear, either in an iPython workbook or better yet a LaTeX document. You may turn in your completed activity the same way you did with CA1 and CA2.

CA1 Solution Sets

Everyone did a nice job on the Class Activities this week. Thank you for all your hard effort and careful work. Coco has placed marked-up PDF files in all of your Dropbox folders. Here are three writeups that I feel constitute the "solution set."

Solution 1 (Peter)
Solution 2 (Melodie)
Solution 3 (Adam)

In the future, please strive to have your write-up be clear enough to serve as the solution set. Science is all about communication, so the clarity of your presentation is a big part of your assessment in this class.

CA1 modification

Based on feedback from Wednesday night's help session, it looks like I underestimated the time necessary to complete the class activity (in true professorial fashion). I forgot that problems 6 and 7 required derivations and that many of you have yet to learn LaTeX, and that problem 8 is nontrivial.

Here's the new plan:

Turn in problems 1-5 at the beginning of class Friday. We'll then spend Friday's class talking about LaTeX and working on fitting lines.

Integrating exponentials

Often in physics, and sometimes in life, you come across the need to integrate an exponential of the form



Allow me to show you how to handle this using simple dimensional analysis rather than calculus and memorization. Dimensional analysis can get you out of a bind when working on a plane (sans wireless), in an oral exam or even during Q&A after your colloquium!

First, note that the units of A must be the same as the units of x since exponentials are dimensionless and dx has units of x. Further, examination of the quantity in the exponent reveals that a must have units of 1/x^2, since the argument of an exponential must be dimensionless, too. Thus, the integral must have units of x and involve a, like so:



This is most of the way there. It turns out that there's a missing factor of the square-root of pi:



But I think it's pretty cool that you can get to within a factor of root-pi (1.77) without any calculus! I can pretty easily remember the pi part after I get the dimensions correct. Even if I forget, being within a factor of two is good enough for astronomy in most applications.

You might notice that this is the form of the Gaussian function, centered on x=0 with



Once normalized, the Gaussian function becomes the normal distribution so frequently used in data analysis (and CA1). Note the distinction between a Gaussian function and a normal distribution. The difference is important, but frequently ignored in the scientific literature. For example, a Gaussian has three free parameters. A normal distribution has only two. And only one of these is a proper probability distribution function (pdf).

For more "Street Fighting Mathematics" like this, check out this book.

Normalization

I'm sorry I wasn't able to be there in class today. However, Coco reports that you all made good progress working on Class Activity 1. She also reports that many of you struggled with the normalization part of Problem (4). Assuming this is where the problem was, allow me to help everyone along.

Problem (4) states:

Without properly normalizing things, you will end up with a proportionality of the form:

In order for this to be an equation, you have to normalize it. You might want to convince yourself that the quantity on the right hand side is not normalized by integrating over all values of \mu. In fact, dividing by this integrated quantity provides the normalization constant:

The denominator is also known as the "evidence":



or the probability of the data given the model, with \mu marginalized out.

The actual value of the integral can be expressed analytically, or you could just do it numerically. Or you can use WolframAlpha. But whatever you do, don't get too hung up on this! :)