Several random variables

Summary

• Given:
• An experiment
• A sample space \(S\) and events
• A probability measure \(P \) assigning a probability to each event. \(P\) satisfies the probability rules.
• We can define \(n\) random variable s \(X_1,…,X_n\) as \(n\) function s , each with domain \(S\) . Each random variable may be discrete, continuous or neither.

Example

• Experiment that picks an outcome from \(S=\{ α,β,γ, θ \}\) . Each outcome is equally likely.
• 16 possible events.
• \(X:S→R\) is defined by \(X\left( α \right)=2\) , \(X\left( β \right)=2\) , \(X\left( γ \right)=2\) and \(X\left( θ \right)=0\) .
• \(Y:S→R\) is defined by \(Y\left( α \right)=0\) , \(Y\left( β \right)=1\) , \(Y\left( γ \right)=1\) and \(Y\left( θ \right)=1\) .

Events

• The expression \(\{ X=x, Y=y \}\) is an event consisting of all outcome that the random variable \(X\) will map into the real number \(x\) and that the random variable \(Y\) will map into the real number \(y\) .

Example (continued)

• \(\{ X=2,Y=1 \}\) is the event \(\{ β,γ \}\) .

Probabilities of events

• \(P(X=x, Y=y)\) is defined as the probability of the event \(\{ X=x,Y=y \}\) .

Example (continued)

\[P\left( X=2,Y=1 \right)=P\left( \{ β,γ \} \right)=1/2\]

• Expressions such as \(\{ X<3, Y<1 \}\) , \(\{ X≥7, Y=2 \}\) and \(\{ -1≤X≤1, Y=2 \}\) are events as well.

Example (continued)

\[P\left( X<2,Y>0 \right)=P\left( \{ θ \} \right)=1/4\]

Several random variables

• The idea is easily extended to \(n\) random variables \(X_1,…,X_n\) where

\[P(X_1=x_1,…, X_n=x_n)\]

• is the probability of the event \(\{ X_1=x_1,…, X_n=x_n \}\) (the collection of outcome where \(X_1=x_1\) and \(X_2=x_2\) and so on)