8 - 18.05 Lecture 8 February 22, 2005 3.1 - Random...

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18.05 Lecture 8 February 22, 2005 § 3.1 -Random Variables and Distributions Transforms the outcome of an experiment into a number. De±nitions: Probability Space: (S, A , P ) S -samp le space, A -events , P -probab i l ity Random variable is a function on S with values in real numbers, X:S R Examples: Toss a coin 10 times, Sample Space = { HTH. ..HT, .... } , all con±gurations of H T. Random Variable X = number of heads, X: S R X: S →{ 0 , 1 , ..., 10 } for this example. There are fewer outcomes than in S, you need to give the distribution of the random variable in order to get the entire picture. Probabilities are therefore given. De±nition: The distribution of a random variable X:S R , is de±ned by: A R , P ( A ) = P ( X A ) = P ( s S : X ( s ) A ) The random variable maps outcomes and probabilities to real numbers. This simpli±es the problem, as you only need to de±ne the mapped R , P , not the
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8 - 18.05 Lecture 8 February 22, 2005 3.1 - Random...

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