# 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)

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