Page 270 - Fundamentals of Probability and Statistics for Engineers

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PROBLEMS
8.1 It has been shown that the frequency diagram gives a graphical representation of the
probability density function. Use the data given in Table 8.1 and construct a diagram
that approximates the probability distribution function of percentage yield X.
8.2 In parts (a)–(l) below, observations or sample values of size n are given for a random
phenomenon.
(i) If not already given, plot the histogram and frequency diagram associated with
the designated random variable X.
(ii) Based on the shape of these diagrams and on your understanding of the
underlying physical situation, suggest one probability distribution (normal,
Poisson, gamma, etc.) that may be appropriate for X. Estimate parameter
value(s) by means of Equations (8.2) and (8.3) and, for the purposes of
comparison, plot the proposed probability density function (pdf) or probabil-
ity mass function (pmf) and superimpose it on the frequency diagram.

(a) X is the maximum annual flood flow of the Feather River at Oroville, CA.
Data given in Table 8.3 are records of maximum flood flows in 1000 cfs for
the years 1902 to 1960 (source: Benjamin and Cornell, 1970).
(b) X is the number of accidents per driver during a six-year time span in
California. Data are given in Table 8.2 for 7842 drivers.
(c) X is the time gap in seconds between cars on a stretch of highway. Table 8.4
gives measurements of time gaps in seconds between successive vehicles at
a given location (n ˆ 100).
(d) X is the sum of two successive gaps in Part (c) above.
(e) X is the number of vehicles arriving per minute at a toll booth on New York
State Thruway. Measurements of 105 one-minute arrivals are given in
Table 8.5.
(f) X is the number of five-minute arrivals in Part (e) above.
(g) X is the amount of yearly snowfall in inches in Buffalo, NY. Given in Table 8.6
are recorded snowfalls in inches from 1909 to 2002.
(h) X is the peak combustion pressure in kPa per cycle. In spark ignition
engines, cylinder pressure during combustion varies from cycle to cycle.
The histogram of peak combustion pressure in kPa is shown in Figure 8.4
for 280 samples (source: Chen and Krieger, 1976).

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