ch 2-Random Variables - STAT 230: Introduction to...

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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 real-valued 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 real-valued 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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ch 2-Random Variables - STAT 230: Introduction to...

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