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Unformatted text preview: Outline Outline 1 Random Variable Random Variables Random Variable Definitions Expectation Variance More properties of R.V.s Outline 1 Random Variable Random Variables Random Variable Definitions Expectation Variance More properties of R.V.s Motivation In the first 2 lectures we talked about rules for probability What wed like to know is how we can use these rules to make some probability statements about observations we make in some experiment To do this we need to add a little bit more structure Random Variables Random Variable Definitions Expectation Variance More properties of R.V.s Random Variable Weve talked about sample spaces and events, but the way weve defined them may not be helpful in analyzing data Imagine we randomize 1000 people to novel a HIVvaccine and 1000 people to a placebo vaccine and want to determine whether there are more HIV infections in the placebo group than the vaccine group S is comprised of elements: e v i = 0 (vaccine, not infected) and e v i = 1 (vaccine, infected) e p i = 0 (placebo, not infected) and e p i = 1 (placebo, infected) S is big : 2 2000 The elements in S arent exactly what we care about either We really care about a function of those elements: X v = 1000 i = 1 e v i and X p = 1000 i = 1 e p i We call X v and X p random variables (R.V.s) Random Variables Random Variable Definitions Expectation Variance More properties of R.V.s Random Variable A R.V. is a function with a domain of S and whose range is a subset of < What? Random Variables Random Variable Definitions Expectation Variance More properties of R.V.s RV Example Random Variables Random Variable Definitions Expectation Variance More properties of R.V.s Random Variable Think of a R.V. as the potential outcome of your experiment number of infections After weve mastered the concept of a RV, well forget everything weve just discussed We wont really talk about sample spaces and events anymore How is it random? In the sense that your experiment is random. R.V.s are always CAPITALIZED (and near the end of the alphabet) Random Variables Random Variable Definitions Expectation Variance More properties of R.V.s R.V. semantics R.V.s can be continuous (blood pressure) or discrete (number of deaths) Were dealing with discrete this week and continuous next week Discrete R.V.s can be countably finite (alive, dead) or countably infinite (0, 1, 2, etc) A R.V. is a possible outcome of an experiment: X An observed outcome is x We typically care about Pr ( X = x ) The probability of observing 10 new HIV infections The probability the number of new HIV infections is larger among the placebo than the among the vaccinated Random Variables Random Variable Definitions Expectation Variance More properties of R.V.s R.V. semantics I mentioned after we start using RVs well stop talking about sample spaces and events However, keep in mind that, formally, when X is a random variable and we say Pr ( X...
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 Spring '11
 RichMacLehose

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