Lecture 04 - 1 Random Variables In applications we are...

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1 Random Variables In applications we are interested in quantitative properties of experimental results. Example: Toss a coin three times and count the number of heads. The sample space is S = { ( t,t,t ) , ( t,t,h ) , ( t,h,t ) , ( h,t,t ) , ( t,h,h ) , ( h,t,h ) , ( h,h,t ) , ( h,h,h ) } . The random variable X counts the number of heads. Thus if w = ( t,t,h ) occurs then X ( t,t,h ) = 1 . Assigning probabilities to value of random variables is done by connecting probability associates with outcomes of an experiment with the values of the random variables. Example: (continued) Assume all elements in S are equally likely. Then X = 1 corresponds to the set { ( t,t,h ) , ( t,h,t ) , ( h,t,t ) } thus P ( X = 1) = P ( { ( t,t,h ) , ( t,h,t ) , ( h,t,t ) } ) = 3 / 8 . Let B = { 2 , 3 } by X B we mean the subset A = { ( t,h,h ) , ( h,t,h ) , ( h,h,t ) , ( h,h,h ) } of S such that X ( w ) B for all w A so P ( X B ) = P ( A ) = 4 8 = 0 . 5 . By X = j we mean the set A = { w S : X ( w ) = j } . So X = 0 is the set A = { ( t,t,t ) } and P ( X = 0) = P ( A ) = 1 / 8 , etc. Definition: A function X : S → < mapping elements of the sample space into the real numbers will be called a random variable . Remark: A random variable is a deterministic function. What is random is the selection of w S . Example: Suppose you select a point at random in the unit circle { ( x,y ) : x 2 + y 2 1 } . Let D be the distance of the selected point to the origin p x 2 + y 2 . Then D is a random variable. 1.1 Discrete Random Variables Definition: Discrete Random Variables: If the possible values of X are finite or countable we say that X is a discrete random variable. Definition: Let x 1 ,x 2 ,... denote the possible values of X then p ( x i ) = P ( X = x i ) = P ( w S : X ( w ) = x i ) is called the probability mass function. Notice that p ( x i ) 0 and that X i =1 p ( x i ) = X i P ( w S : X ( w ) = x i ) = P ( S ) = 1 . Convenient notation (not used in book): p X ( a ) = P ( X = a ). This notation is useful when you want to emphasize that you are refereing to random variable X . When there is no danger of confusion we drop the subscript. Example: (continued) p (0) = 1 / 8, p (1) = 3 / 8 , p (2) = 3 / 8, p (3) = 1 / 8. The probability mass function summarizes all probability information associated with the random vari- able. 1
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Example: (continued) P ( X 1) = p (0) + p (1) = 0 . 5 . P (1 X 2) = p (1) + p (2) = 3 / 4, P ( X > 1) = p (2) + p (3) = 0 . 5. Although the probability mass function contains all information w.r.t. a discrete random variable, the cumulative distribution function is frequently used. Definition: The cumulative distribution function (cdf) F of a random variable X is given by F X ( a ) = P ( X a ) .
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This note was uploaded on 06/02/2010 for the course IEOR SIEO W4150 taught by Professor Gallego during the Spring '04 term at Columbia.

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Lecture 04 - 1 Random Variables In applications we are...

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