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Unformatted text preview: Example 2.8: 0.1172. The contour of the
histogram converges to a
smooth curve f (x). The
area becomes an integral:
185
140 160
180
height (cm) f (x) dx 200 180 igure 2.16: The total area under the curve representing all individuals is 1. The
rea between 180 and 185 cm is the fraction of the US adult population between ORF 245 Prof. Rigollet
Fall 2012 Probability density function
The function f (x) is called probability density
function (pdf ) of the continuous random variable X.
In our example, X is the (random) height of a randomly
selected US adult.
68 2.2.2 CHAPTER 2. PROBABILI Probabilities from continuous distributions We computed the proportion of individuals with heights 180 to 185 cm in Exa
ple 2.8 as a proportion:
number of people between 180 and 185
total sample size We found the number of people with heights between 180 and 185 cm by de
mining the shaded boxes in this range, which represented the fraction of the
area in this region.
Similarly, we use the area in the shaded region under the curve to ﬁn
probability (with the help of a computer): We use it to compute the probability that X is in a given
interval [a,b]: P (height between 180 and 185) = area between 180 and 185 = 0.1157 The probability a randomly selected person is between 180 and 185 cm is 0.11
This is very close to the estimate from Example 2.8: 0.1172. P (a X b b) = f (x)dx a 140 160
180
height (cm) 200 In particular, the total area under the curve of f (x) is 1. Figure 2.16: The total area under the curve representing all individuals is 1. T
area between 180 and 185 cm is the fraction of the US adult population betw
180 and 185 cm. Compare this plot with Figure 2.14. total area = ⇥
⇥ Exercise 2.20 Three US adults are randomly selected. The probability a si
adult is between 180 and 185 cm is 0.1157. Short answers in the footnote20 .
(a) What is the probability that all three are between 180 and 185 cm tall? f (x)dx = P ( ⇤ ⇥ X ⇥ ⇤) = 1 (b) What is the probability that none are between 180 and 185 cm?
Exercise 2.21 What is the probability a randomly selected person is exactly
cm? Assume you can measure perfectly. Answer in the footnote21 . 20 (a) 0.1157 ⇥ 0.1157 ⇥ 0.1157 = 0.0015. (b) (1 0.1157)3 = 0.692
This probability is zero. While the person might be close to 180 cm, the probability th
randomly selected person is exactly 180 cm is zero. This also makes sense with the deﬁnitio
probabili...
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 Fall '09

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