Probability (statistics square one)

I put all the blame on my procrastination! It has been so long since I wanted to know what is Kalman filter and how to use it, but all the fun I had with my games hinders me to do so. To cut it short, I did some browsing on the net and found that Kalman filter is actually based on Bayesian inference, which is a statistical method. Knowing that, I was both happy and sad. I was happy because now I know one of the uses of statistics which I learned years ago when I was sitting in undergraduate school. The sad thing is I already forgot them all! So, in this post I will share what I learned about statistics, from square one. I hope you will get something from it.

In most of the statistics books I glance through, they always have probability as one of the topics. So I will start with learning what it actually is. Let $x$ be the set of the possible outcomes, and $F$ is a collection of subsets of $x$. A probability on $(x, F)$ is $\mu : F\rightarrow [0, 1]$. In other words, to every set in $F$, $\mu$ assigns a probability between 0 and 1. $\mu$ will be called as set function because its domain is a collection of sets. To be a probability $\mu$ must:

1. The result of $\mu (\varnothing )=0$, where $\mu$ is the empty set,
2. The result of $\mu (x)=1$,
3. if $A_1$ and $A_2$ are disjoint, then $\mu (A_1 \cup A_2) = \mu(A_1) + \mu(A_2)$.

This will be easier to grasp if we consider a six-sided die as an example. The set of outcomes (which is $x$) is $\left \{ 1, 2, 3, 4, 5, 6 \right \}$. If we believe the die to be fair we could assign $\mu (1) = \mu (2) = \mu (3) = \mu (4) = \mu (5) = \mu (6) = 1/6$. The third point above imply various other values such as $\mu (1, 2) = 1/3$ and $\mu (3, 5, 6) = 1/2$. Now those three points above make sense.

It is important to remember that setting $\mu (i) = 1/6$ is not simply because the die has six faces. We set $\mu (i) = 1/6$ because we believe the die to be fair. It seems in probability there are a lot of things to consider.

Following phrases will be used interchangeably when we talk about probability:

1. The probability that die lands $1$,
2. Can be written as $P(1)$,
3. Can be written as $P[the die lands 1]$,
4. Can be written as $\mu ({1})$,
5. Can be written as $\mu (1)$.

We also use the word distribution in place of probability or probability measure.

The next article will discuss about the density of probability. But before that, I need to refresh my brain about basic mathematical notations, since most of the references I read used tons of them.