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# 04 013 025 039 017 002 e x 2 004 3 013

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Unformatted text preview: ) = 2 · 0.04 + 3 · 0.13 + 4 · 0.25 + 5 · 0.39 + 6 · 0.17 + 7 · 0.02 = 4.58 var(X ) = 22 · 0.04 + 32 · 0.13 + 42 · 0.25 + 52 · 0.39 + 62 · 0.17 + 72 · 0.02 (4.58)2 = 1.20 ORF 245 Prof. Rigollet Fall 2012 Continuous random variables Consider the histogram of heights of 3 million US adults: 2. CONTINUOUS DISTRIBUTIONS 140 160 height (cm) 180 67 200 gure 2.14: A is the probability that aof 2.5 cm. The shaded region represents What histogram with bin sizes randomly selected US adult dividuals with heights between 180 and 185 cm. has height between 180 and 185 cm? .2.2 Probabilities from continuous distributions e computed the proportion of individuals with heights 180 to 185 cm in Examle 2.8 as a proportion: Continuous random variables number of people between 180 and 185 total sample size e found the number of people with heights between 180 and 185 cm by deterining the shaded boxes in this range, which represented the fraction of the box ea in this region. Similarly, we use the area in the shaded region under the curve to ﬁnd a robability (with the help of a computer): It is given by the area in orange (sum of 2 rectangles) P (height between 180 and 185) = area between 180 and 185 = 0.1157 What if we let the bin width go to 0? he probability a randomly selected person is between 180 and 185 cm is 0.1157. his is very close to the estimate from Example 2.8: 0.1172. 140 160 180 height (cm) 200 igure 2.16: The total area under the curve representing all individuals is 1. The ea between 180 and 185 cm is the fraction of the US adult population between ORF 245 Prof. Rigollet Fall 2012 8 CHAPTER 2. PROBABILITY .2.2 ORF 245 Prof. Rigollet Fall 2012 Probabilities from continuous distributions e computed the proportion of individuals with heights 180 to 185 cm in Examle 2.8 as a proportion: Continuous random variables number of people between 180 and 185 total sample size e found the number of people with heights between 180 and 185 cm by deterining the shaded boxes in this range, which represented the fraction of the box rea in this region. Similarly, we use the area in the shaded region under the curve to ﬁnd a robability (with the help of a computer): It is given by the area in orange (sum of 2 rectangles) P (height between 180 and 185) = area between 180 and 185 = 0.1157 What if we let the bin width go to 0? he probability a randomly selected person is between 180 and 185 cm is 0.1157. his is very close to the estimate from...
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