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Rigollet fall 2012 in statistics we think of our

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Unformatted text preview: ract 6. The gas milage of a r.s. car made in 2010 Random variables ORF 245 Prof. Rigollet Fall 2012 In statistics, we think of our (numerical) data as the realization of random variables (after the random experiment has taken place). For example, the list of numbers in medv are realizations of the random variable MEDV. Note that we typically use uppercase for random variables and lowercase for the realizations. Random variables can also be split into: discrete and continuous ORF 245 Prof. Rigollet Fall 2012 0.5 Random variables 0.4 discrete Takes values 0, 1, 2, .... 0.1 0.2 0.3 Completely described by P(X=1), P(X=2), .... 0.0 x=c(1, 1, 1,1,1,1,1,1,1,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3) barplot(table(x))able(x))) 1 2 3 continuous Takes a continuum of values (e.g. interval [a, b]) Can be described by the limiting histogram when the bin size goes to 0. CHAPTER 2. PROBABILITY CHAPTER 2. PROBABILITY probability 66 probability 66 Four histograms of US adults heights with varying bin widths. 140 160 160 180 180 200 140 160 160 180 180 200 200 140 140 160 160 180 180 200 height (cm) height (cm) robability height (cm) height (cm) 140 140 160 160 180 180 200 200 200 140 height (cm) height (cm) probability probability 140 140 height (cm) height (cm) 200 140 160 160 180 180 200 200 140 140 160 160 180 180 200 200 height (cm) height (cm) probability CHAPTER 2. PROBABILITY CHAPTER 2. PROBABILITY probability 66 robability 66 height (cm) height (cm) Figure FigureFour histograms of US adults adults heights with varying bin widths. 2.13: 2.13: Four histograms of US heights with varying bin widths. Discrete random variables ORF 245 Prof. Rigollet Fall 2012 To fully describe discrete random variables, we need to give P(X=x), for each value x that the random variable X can take. Examples: Possible values for X 1. Rolling one die: {1,2,...,6} 2. Sum of two dice: {1,2,...,12} 3. Number of cars in household: {0,1, 2, ....} 4. Indicator of sickness: {0, 1} 5. Number of courses this semester: {2, ...,7} ORF 245 Prof. Rigollet Fall 2012 Probability distribution The probability distribution of a discrete random variable X is a table (or a formula) that gives P(X=x) for each possible value x of X. x 2 3 4 5 6 7 P(X=x) 0.04 0.13 0.25 0.39 0.17 0.02 Number of courses ORF 245 Prof. Rigollet Fall 2012 Probability distribution It is equivalent to add values x for X that are not possible and assign them probability 0 x 0 1 2 3 4 5 6 7 P(X=x) 0.00 0.00 0.04 0.13...
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