# Random Variables

## 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: $P(f=n) = P(\{a | F(a)=r\})$

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: $0 ≤ P(f=r) ≤ 1$ $Σ P(f=r) = 1$ 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 $¬(F=r) = \{a | f(a) ≠ r\}$ $P(¬(F=r)) = 1 - P(F=r)$ $P(f=r_1) V P(f=r_2) = P(f=r_1) + P(f=1_2) - P(f=r_1, f=r_2)$ Where “,” is used as “and” & “and” is used as “intersection”

## Conditional Probability

If p(F_2 = r_2) does not = 0 then: $P(f_1=r_1 | f_2=r_2) = \frac{p(f_1=r_1, f_2=r_2)}{p(f_2=r_2)}$

The multiplication rule is also applicable to random variables $P(f_1=r_1, f_2=r_2) = P(f_1=r_1 | f_2=r_2) * P(f_2=r_2)$

We sometimes use symbols distinct from numbers to represent the value of a random variable. Like F(weather = sunny).

## Probability distrubtion

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:

$P(weather=sunny)=0.7$ $P(weather=rain)=0.2$ $P(weather=cloudy)=0.08$ $P(weather=Storm)=0.02$

### 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.

## Marginalisation

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)

## Probabilist Inference

Can be charecterised as the computation of potential probabilities $P(F|E_1 = e_1,..., E_m = e_m)$ 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.

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