Financial Mathematics: Statistics - Expected Values

In statistics, a probability distribution is any function, $f(x)$ which is never negative (probability is either 0 or positive) and it sums up to 1 (100%). Mathematically we'd express this as:
\[\begin{align*}
\forall x f(x)\ge 0 \\
\sum_x f(x) = 1
\end{align*}\]
In the case of a continuous random variable, the second formula would be expressed as an integral:
$$\int_x f(x)dx = 1$$
There are some differences between discrete and continuous random variables but the ideas behind them are the same.

To motivate the idea behind an expected value, we'll begin with a more familiar concept: an average.

The Efficient Markets Hypothesis is Meaningless

I'm going to begin a critique of the Efficient Markets Hypothesis (EMH). This is not the first nor will it be the last that have been presented. Most of these critiques accept the basic paradigm an attempt to empirically prove that one can "beat the market".

For the practitioner, this can be quite appealing as it allows one to find some strategy that would allow one to earn "excess returns".

My approach, which I've been toying with in my mind for the last year or so, is going to be a bit different. My contention is that the entire paradigm is questionable and perhaps "meaningless"¹. At the very least, proponents of EMH have a a lot more work to do as there is a lot of ideological baggage and not much in the way of a legitimate scientific hypothesis.

How to Ignore the Noise in Financial News

One of the most difficult things we face in the information age is the problem of too much information. It's everywhere around us. There's absolutely no way for us to get through all of that information much less be able to utilize it.

There's even a good deal of research that indicates that, not only are we unable to handle extra information, that additional information may make us less accurate and more confident in our inaccurate predictions: The illusion of knowledge: When more information reduces accuracy and increases confidence.

Now it seems to me that there are at least three goals we need to focus on in order to handle all of this information:

1. Focus on important information.
2. Ignore the useless noise.
3. Know what we do not know.

While I think this is important in general terms, there is the question of how to deal with financial news.