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Unformatted text preview: STAT 230: Introduction to Probability and Random Variables Summer 2009 Chapter 2 1 Random Variables For many experiments, we are more interested in a characteristic of the outcome rather than the outcome itself. The notion of a variable is introduced in elemen tary algebra courses. Such a concept extends to the situation in which the values taken by a variable are determined by a random mechanism or experiment. Definition 1. Random Variable A random variable is a realvalued function whose domain is the sample space of a random experiment; i.e. X : Ω → < . That is, for any simple outcome ω ∈ Ω , we can take a realvalued measurement X ( ω ) on this outcome. Example 1. Let Ω be the sample space of outcomes when a fair die is rolled twice and let X be the sum of the two resulting faces. Then X(3, 4) = 7, X(1, 2) = 3, etc. Example 2. Let Ω be the sample space of outcomes when a fair coin is tossed and let X be the indicator of getting a ”Head”: X ( ω ) = 1 if ω = ”Head”, X ( ω ) = 0 if ω = ”Tail” We are interested in the probability of events that are defined by random variable, such for example the probability that the sum of two rolls of a fair die is even, etc. We define A = { ω ∈ Ω : X ( ω ) ∈ K } , we are interested in computing P ( A ) = P ( { ω ∈ Ω : X ( ω ) ∈ K } ) We can use the shorthand notation A = { X ∈ K } , then P ( A ) = P ( X ∈ K ) . 1 Example 3. If Ω consists of the outcomes of two rolls of a die and X is the sum, then the event that the sum is even can be written as A = { ω ∈ Ω : X ( ω ) ∈ { 2 , 4 , 6 , 8 , 10 , 12 }} and the probability is P ( A ) = P ( { X ∈ { 2 , 4 , 6 , 8 , 10 , 12 }} ) = 1 2 . We will study two kinds of random variables: discrete and continuous. Dis crete random variables can be thought of as counts taking integer values. Con tinuous random variables can be thought of measurements taking real values. Definition 2. Discrete Random Variable A random variable X is called discrete if there is a countable set K ⊂ < such that P ( { X ∈ K } ) = 1 . Remark The adjective countable is a mathematical term that means you are able to count all the points in K. A countable set is not necessarily finite !! For example, the set of natural numbers is countable but infinite. Example 4. Let X denote the number of Heads when a coin is tossed three times. The sample space is : Ω = { HHH,HHT,HTH,THH,HTT,THT,TTH,TTT } ω HHH HHT HTH THH HTT THT TTH TTT X ( ω ) 3 2 2 2 1 1 1 Then, P ( X = 2) = P ( { HHT,HTH,THH } ) = 3 8 ....
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