Introduction of Random Variables
Let $(S,P)$ be a probability space. A random variable $F$ is a function $F:S\rightarrow\Bbb{R}$ that assigns to every $s\in S$ a single number $F(s)$.
In all these examples $S$ is the sample space and $P$ is the probability space.
- Neither a variable nor random.
- As it is the product of a function.
- English translation of variable casuale.
We still assume that the sample space is finite. Thus, given a random variable $F$ from some sample space $S$, the set of number $r$ that values of $F$ is finite as well.
The event that $F$ takes the value $r$, that is $\{s\vert F(s)=r\}$, is denoted $(F=r)$. the probability $(F=r)$ of the even $(F=r)$ is then:
\[P(F=r)=P(\{s\vert F(s)=r\})\]Example 1
Let
\[S=\{\text{car, train, plane, ship}\}\]Then the function $F:S\rightarrow \Bbb{R}$ defined by:
\[\begin{aligned} F(\text{car})&=1\\ F(\text{train})&=1\\ F(\text{plane})&=2\\ F(\text{ship})&=2 \end{aligned}\]is a random variable.
$(F=1)$ denotes the event $\{s\in S \vert F(s) =1\} = \{\text{car, train}\}$.
Defining a uniform probability space $(S,P)$ by setting:
\[P(\text{car})=P(\text{train})=P(\text{plane})=P(\text{ship})=\frac{1}{4}\]This means that we are setting each event to have the same probability.
Then:
\[\begin{aligned} P(F=1)=&P(\{s\in S \vert F(s)=1\})\\ =&P(\{\text{car, train}\})\\ =&\frac{1}{2} \end{aligned}\]Example 2
Suppose that I roll two dice. so the same sample space is:
\[S=\{1,2,3,4,5,6\}^2\]and $P(ab)=\frac{1}{36}$ for every $ab\in S$.
Let
\[F(ab) = a+b\]This means that we are investigating the sum of the two dice.
$F$ is a random variable. This variable maps the two values $ab$ to the sum $a+b$.
The probability that $F=r$ for a number $r$ (say, 12) is given by:
\[P(F=r)=P(\{ab\vert F(ab)=r\})\]With this example:
\[\begin{aligned} P(F=12)=&P(\{ab\vert F(ab)=12\})\\ =&P(66)\\ =&\frac{1}{36} \end{aligned}\]Random Variables
when defining a probability distribution $P$ for a random variable $F$, we often do not specify its sample space $S$ by directly assign a probability to the event that $F$ takes a certain value. Thus we directly define the probability:
\[P(F=r)\]of the event that $F$ has the value $r$. Observe:
- $0\leq P(F=r)\leq 1$
- $\sum_{r\in \Bbb{R}}P(F=r)=1$
Thus, the events $(F=r)$ behave in the same way as outcomes of a random experiment.
Notation and Rules
Negation
We write $\neg(F=r)$ for the event $\{s\vert F(s)\neq r\}$. For example, assume the random variable $\text{Die}$ can take values $\{1,2,3,4,5,6\}$ and:
\[P(\text{Die}=n)=\frac{1}{6}\]for all $n\in \{1,2,3,4,5,6\}$ (thus we have a fair die).
The $\neg(\text{Die}=1)$ denotes the event:
\[\begin{aligned} &(\text{Die}=2)\vee (\text{Die}=3)\vee (\text{Die}=4)\vee\\ &(\text{Die}=5)\vee (\text{Die}=6) \end{aligned}\]We have the following complementation rule:
\[P(\neg(F=r))=1-P(F=r)\]Intersection
We can write:
\[(F_1=r_1,F_2=r_2)\]for $(F_1=r_1)$ and $(F_2=r_2)$. This takes two random variables
Union
Two represent the function OR we can write:
\[(F_1=r_1)\vee(F_2=r_2)\]From this we can get the law from unions:
\[\begin{aligned} P((r_1)\vee (r_2)) =& P(r_1)+P(r_2)\\ &-P(r_1,r_2) \end{aligned}\]Conditional Probability
if $P(F_2=r_2)\neq0$ then:
\[P(F_1=r_1\vert F_2=r_2)=\frac{P(F_1=r_1,F_2=r_2)}{P(F_2=r_2)}\]This is the probability that $F_1=r_1$ given that $F_2=r_2$.
Product Rule
\(\begin{aligned} P(F_1=r_1,F_2=r_2)=&P(F_1=r_1\vert F_2=r_2)\\ &\times P(F_2=r_2) \end{aligned}\)
Symbols
Wo sometimes use symbols distinct from numbers to denote the values of random variables.
For example, for a random variable $\text{Weather}$ rather than using values $1,2,3,4$ we use:
\[\text{sunny, rain, cloudy, snow}\]Thus, $(\text{Weather}=\text{sunny})$ denotes the event that it is sunny.
This is to say that random variable must give numbers as an output but we use symbols to represent these numbers.
To model a visit to a dentist, we use the random variables $\text{Toothache, Cavity}$ and $\text{Catch}$ (the dentist’s steel probe catches in the tooth) that all the take values 1 and 0 (for True or False).
For example, ($\text{Toothache}=1$) states that the person has toothache and ($\text{Toothache}=0$) states that the person does not have a toothache.