Page 269 - Fundamentals of Probability and Statistics for Engineers

252 Fundamentals of Probability and Statistics for Engineers

evaluated in a subjective way by using the frequency diagram given in Figure 8.1.
The normal density function with mean 70 and variance 4 is superimposed on the
frequency diagram in Figure 8.1, which shows a reasonable match. Based on this
normal distribution, we can calculate the probabilities given above, giving a further
assessment of the adequacy of the model. For example, with the aid of Table A.3,

68 70
P X < 68†ˆ F U ˆ F U 1†
2
ˆ 1 F U 1†
ˆ 0:159;

which compares with 0.13 with use of the frequency diagram.
In the above, the choice of 70 and 4, respectively, as estimates of the mean
and variance of X is made by observing that the mean of the distribution should
be close to the arithmetic mean of the sample, that is,

n
1 X
m X  x j ; 8:2†
n
jˆ1
and the variance can be approximated by
n
2
X
2
 1 x j m X † ; 8:3†
X n
jˆ1
which gives the arithmetic average of the squares of sample values with respect
to their arithmetic mean.
Let us emphasize that our use of Equations (8.2) and (8.3) is guided largely
by intuition. It is clear that we need to address the problem of estimating the param-
eter values in an objective and more systematic fashion. In addition, procedures
need to be developed that permit us to assess the adequacy of the normal model
chosen for this example. These are subjects of discussion in the chapters to follow.

REFERENCES

Benjamin, J.R., and Cornell, C.A., 1970, Probability, Statistics, and Decision for Civil
Engineers, McGraw-Hill, New York.
Burg, A., 1967, 1968, The Relationship between Vision Test Scores and Driving Record,

two volumes. Department of Engineering, UCLA, Los Angeles, CA.
Chen, K.K., and Krieger, R.R., 1976, ‘‘A Statistical Analysis of the Influence of Cyclic
Variation on the Formation of Nitric Oxide in Spark Ignition Engines’’, Combustion

Sci. Tech. 12 125–134.

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