Hacker School, Monday, July 7th, 2014

I spent most of the weekend working on moving my blog off of Blogger. I imported it to Nikola, a static site generator, and spent many hours figuring out how to configure it to have both a main blog for articles, and a journal blog in the background. I also had to learn how to host it on GitHub. I subsequently wrote two(!) posts about how that process went.

This blog moving used up time I had allocated to Dan Boneh's Coursera cryptography course, and so I still had that to do on Monday. Not only that, but I spent more time trying to configure more blog things for the first few hours of the day.

I did finally get to the cryptography, though. I'm pleased about that, because I love cryptography and coding theory, and have been wanting a refresher for a long time. I'd like to turn some of my notes into a more formal post, but there was one small point that I'd like to mention.

The term random variable has long bothered me. I don't find the concept confusing, but the name for it makes it hard for me to remember what it is. A random variable is just a function that maps a set to a probability distribution. A randomised algorithm is defined here in essentially the same way.

I guess I would prefer if the variable were abstracted away entirely.


The above statement "A random variable is just a function that maps a set to a probability distribution." is not technically correct.

A random variable is a function from the universe to some outcome set. However, the random variable implicitly induces a probability distribution on the outcome set based on the distribution of the outputs given a uniform distribution on the inputs. That is, if the random variable is \(X:U \to V\), then the probability distribution that \( X \) induces on \( V \) can be described by a function \(F: V \to R\), where \(F(v) = P(X^{-1}(v))\). This is usually written as \(P(X=v)\).

The idea I was trying to convey was that the F form of the function is a kind of pointfree function, that feels more intuitive to me. Something like this:

\(F: V \to R = lambda X: P(X^{-1}(v))\)

I'm not sure how much sense this makes, either. I confess I'm a bit underslept.


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