ABSTRACT
We are now in a position to define a random variable. In a more elementary probability course, you may have learned that a random variable X = X(ω) is a real-valued function whose domain is Ω. For instance, in Example 3.4, where each ω is a specific set of 5 cards, let X(ω) = I{ω contains four aces}take the value 1 if ω contains 4 aces and 0 otherwise. Then X(ω) is a random variable. We calculate the probability of an event involving X by using the P measure of the set of ω that get mapped to the X event. In this example, the probability that X = 1 is just P {Ace of Diamonds, Ace of Hearts, Ace of Clubs, Ace of Spades, 2 of Diamonds} + P {Ace of Diamonds, Ace of Hearts, Ace of Clubs, Ace of Spades, 2 of Hearts} + … + P {Ace of Diamonds, Ace of Hearts, Ace of Clubs, Ace of Spades, King of Spades} = 48/ ( 52 5 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315370576/0f35c063-5312-4158-98aa-e82231747ad7/content/eq97.tif"/> because each of the 48 disjoint simple events comprising I{ω contains four aces} = 1 has probability 1/ ( 52 5 ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315370576/0f35c063-5312-4158-98aa-e82231747ad7/content/eq98.tif"/>
