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10-07 Random Variables (1)

10-07 Random Variables (1) - BTRY 4080 STSCI 4080 Fall 2010...

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BTRY 4080 / STSCI 4080 Fall 2010 175 Chapter 4: Random Variables Definitions: A random variable is a function from the sample space S to the real numbers. A discrete random variable is a random variable that takes on (at most) a countable number of values. A continuous random variable is a random variable that takes values on a continuum of possible values. Focus of Chapter 4: discrete random variables.

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BTRY 4080 / STSCI 4080 Fall 2010 176 Example (see also 4.1.1a): Suppose the experiment consists of tossing 3 coins. Sample space: S = ( s 1 = ( H,H,H ) s 2 = ( H,T,T ) s 3 = ( T,H,T ) s 4 = ( T,T,H ) s 5 = ( H,H,T ) s 6 = ( H,T,H ) s 7 = ( T,H,H ) s 8 = ( T,T,T ) ) Let Y denote the number of heads. Then, Y is a discrete random variable taking on one of the values 0, 1, 2, 3 . Formally: Y is a function from S to the real numbers (i.e., a set function): Y ( s 1 ) = 3 Y ( s 2 s 3 s 4 ) = 1 Y ( s 5 s 6 s 7 ) = 2 Y ( s 8 ) = 0
BTRY 4080 / STSCI 4080 Fall 2010 177 We can assign probabilities using what we know about the probabilities of each event. If P ( H ) = p and tosses are independent, then: P ( Y = 0) = P ( s 8 ) = (1 p ) 3 P ( Y = 1) = P ( s 2 ) + P ( s 3 ) + P ( s 4 ) = 3 p (1 p ) 2 P ( Y = 2) = P ( s 5 ) + P ( s 6 ) + P ( s 7 ) = 3 p 2 (1 p ) P ( Y = 3) = P ( s 1 ) = p 3 Easy algebra shows that P parenleftBigg 3 uniondisplay i =0 { Y = i } parenrightBigg = 3 summationdisplay i =0 P { Y = i } = 1 .

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BTRY 4080 / STSCI 4080 Fall 2010 178 Some important features: Each s i is assigned to one, and only one, value of Y ; however, more than one s i can be assigned to a given value of Y . Put somewhat differently: Y ( · ) , regarded as a function from S to R , is onto but not necessarily one-to-one . If i negationslash = j , then { Y = i } and { Y = j } correspond to mutually exclusive subsets of outcomes in S ; Original probability structure defined on S
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10-07 Random Variables (1) - BTRY 4080 STSCI 4080 Fall 2010...

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