Random variable and probabilities

Summary

Notation {X = x}

• Given:
• An experiment
• A sample space \(S\) and events
• A probability measure \(P \) assigning a probability to each event. \(P\) satisfies the probability rules.
• A random variable \(X\)
• The expression \(\{ X=x \}\) is an event consisting of all outcome that the random variable \(X\) will map into the real number \(x\) .

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 \(X\left( α \right)=1\) , \(X\left( β \right)=2\) , \(X\left( γ \right)=3\) and \(X\left( θ \right)=3\) .
• \(\{ X=3 \}\) is the same as \(\{ γ, θ \}\) .

Notation P(X = x)

• \(P(X=x)\) is defined as the probability of the event \(\{ X=3 \}\)

Example (continued)

\[P\left( X=3 \right)=P\left( \{ γ, θ \} \right)=1/2\]

• Expressions such as \(\{ X<3 \}\) , \({X≥7}\) and \({-1≤X≤1}\) are events as well.
• \({X<3}\) is the collection of all outcome that the random variable \(X\) will map into a number less than 3.

Example (continued)

\[P\left( X<3 \right)=P\left( \{ α,β \} \right)=1/2\]