1.4 Primer in statistics

1.4.1 Random variables(확률 변수)

1.2 Random Variables MED INTRO V2-en

When we do nonlinear filtering,

we need them to describe the quantity that we're interested in, for example, the position of a vehicle.
We also need random variables to describe the observations that we want to filter.

To describe our random variables, we'd use the

so-called probability mass function(PMF) for discrete-valued random variables and a
so-called probability density function(PDF) for continuous valued random variables.

표기법 : Now, the probability mass function of a discrete-valued random variable is, in this course, denoted as Pr of z, or just P of z.

Note also that we are using braces here to indicate that z is a discrete-valued random variable.

Now, our probability mass function need to have the following properties in order to be proper probability mass functions.

특징 #1 : First, the probability that our discrete-valued random variable z is equal to some integral value i, which is written like this, needs to be greater than or equal to 0.
Now, one way to view this value here is if we collect many values of z, the fraction of these that are equal to i is given by this number here.
And this needs to hold for all values of i.
특징 #2 : The second property of our probability mass function is that if we sum over all values of z, this sum needs to be 1.
That is, the probability that z takes any value needs to be 1.
We can also note that as a consequence of these two, we cannot have a probability mass for a value i that is greater than 1, which seems to be reasonable, right?

예 : Now if we look at this use in the example of a fair dice, the probability mass function for the face value that I would get if we rolled the dice can be written like this.

So the dice has six faces with a value 1 through 6, which each is equally probable.
So the probability that z is i is equal to 1/6, if i is equal to 1, 2, and so on up to 6, and 0 otherwise.
If we visualize this pmf, it will look something like this, where we only have probability mass for discrete values.