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Unformatted text preview: 1 8 1 6 1 4 1 2 1 0 8 6 4 2 8 0 0 7 0 0 6 0 0 5 0 0 4 0 0 3 0 0 2 0 0 1 0 0 N u m b e r o f p a p e r s Frequency Problem 1.8 (a) The histogram is: The most interesting feature of the histogram is the heavy positive skewness of the data. (b) From the frequency distribution (or from the histogram), the number of authors who published at least 5 papers is 33+28+19+…+5+3+3 = 144, so the proportion who published 5 or more papers is 144/1309 = .11, or 11%. Similarly, by adding frequencies and dividing by n = 1309, the proportion who published 10 or more papers is 39/1309 = .0298, or about 3%. The proportion who published more than 10 papers (i.e., 11 or more) is 32/1309 = .0245, or about 2.5%. Problem 1.16 A histogram of the raw data appears below: 80 70 60 50 40 30 20 10 15 10 5 ID T va lue Frequency After transforming the data by taking logarithms (base 10), a histogram of the log 10 data is shown below. The shape of this histogram is much less skewed than the histogram of the original data. Problem 1.20 (a) The density function is 10 / 1 )] 5 ( 5 /[ 1 ) ( = = x f over the interval [5, 5] and ) ( = x f elsewhere. The proportion of x values that are negative is exactly .5 since the value = x sits precisely in the middle of the interval [5, 5]. (b) The proportion of values between 2 and 2 is )] 2 ( 2 [ 10 1 = .4. The proportion of the x values falling between 2 and 3 is )] 2 ( 3 [ 10 1 = .5. (c) The proportion of the x values that lie between k and 4 + k is ] ) 4 [( 10 1 k k + = .4. Problem 1.23 (a) x λ = .00004 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 9 8 7 6 5 4 3 2 1 logarithm of IDT Frequency (b) dx e x ∫ ∞ 0000 , 20 00004 . 00004 . = ∞  000 , 20 00004 . ) 00004 (. ) 00004 (. x e = ∞ 20,000 00004 . x e = 0  (e.00004(20,000) ) = e.8 = .449. Note: for any exponential density curve, the area to the right of some fixed constant always equals e λ c , as our integration above shows. That is, Proportion (x >c) = ∫ ∞ c x dx e λ λ = e λ c . Proportion (x ≤ 30,000) = 1  Proportion (x > 30,000) = 1  e λ c = 1  e.00004(30,000) = 1  e1.2 = .699. Proportion (20,000 ≤ x ≤ 30,000) = Proportion (x > 30,000)  Proportion(x ≤ 20,000) = .699  (1.449) = .148....
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 Spring '08
 Probability distribution, probability density function, Logarithm

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