What is a Random Variable?
A random variable is not a variable or random. It is a function that maps the output to the real numbers.
We will assume that the sample space is finite. Thus, given a random variable, F, from a sample space S, the set of numbers n that take the values of F is finite as well.
The probability that F takes the value N, in symbols (F=N), is defined as:
When defining a probability distribution P for a random variable F we often do not apply it’s sample space S but directly assign a probability to the event that F takes a certain value.
Thus we define the probability P(f=r) of the event that F has value R as: This is just basic probability. The probability of one single random variable is between 0 and 1. The sum of all random variables is 1.
Notation and rules
We write Where “,” is used as “and” & “and” is used as “intersection”
If p(F_2 = r_2) does not = 0 then:
The multiplication rule is also applicable to random variables
We sometimes use symbols distinct from numbers to represent the value of a random variable. Like F(weather = sunny).
The probability distrubtion for a random variable gives the probabilities of all the possible values of the variable. Assume the order of the variables is fixed then:
Joint Probability Distribution
Let f1,…,fk be random variables then a joint probability distribution for them gives the probabilities P(f1=r1,…,fk=rk) for a domain of interest.
Full Joint Probability Distribution
A full joint probability distribution is a joint probability distribution for all relevant random variables f1,…,fk for a domain of interest.
Every probability question about a domain can be answered by the full joint probability distrubtion because the probabilirty of any event is a number of probabillities.
Note: n1…nk are often called data points or sample points.
A full joint probability distrubtion will only have information about a domain of interest. A non-full distrubtion could contain information about a domain you don’t care about.
Given a joint distribution P(f1,…,Fk), one can compute the unconditional on marginal probabillities of the random variables Fi by summing out the remaining values.
Conditional / Posterior distrubitions
We can also compute conditional / posterior distributions from the full joint distribution. We use the P notation for conditional distributions.
|P(F||G) gives the conditional / postieor distrubtion of F given G given by the probabilities P(f=r||G=s) for all values r and s.|
Using this notation, the general version of the multiplication / product rule is:
|P(F, G) = P(F||G)P(G)|
Can be charecterised as the computation of potential probabilities For every variables F given derived evidence E_1,…,E_2.
The denominator can be viewed as a marginalation constant for the distrubtion P, ensuring that it adds up to 1.