Page 220 - Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 203

and 11.2 cm, what proportion of the manufactured parts will be rejected on
average?
Answer: If X is used to denote the part length in centimeters, it is reasonable
to assume that it is distributed according to N(11, 0.04). Thus, on average, the
proportion of acceptable parts is P"10:6 < X 11:2). From Equation (7.25),
and using Table A.3, we have

10:6 11 11:2 11

P 10:6 < X 11:2†ˆ P < U
0:2 0:2
ˆ P 2 < U 1†ˆ F U 1† ‰1 F U 2†Š

ˆ 0:8413 1 0:9772†ˆ 0:8185:

The desired answer is then 1 0:8185, which gives 0.1815.
The use of the normal distribution in Example 7.3 raises an immediate
concern. Normal random variables assume values in positive and negative
ranges, whereas the length of a machine part as well as many other physical
quantities cannot take negative values. However, from a modeling point of
view, it is a commonly accepted practice that normal random variables are valid
representations for nonnegative quantities in as much as probability P(X < 0)
is sufficiently small. In Example 7.3, for example, this probability is

11

P X < 0†ˆ PU < ˆ P U < 55† 0
0:2

Example 7.4. Let us compute P"m k < X m ‡ k ) where X is distrib-

uted N(m, 2 ). It follows from Equations (7.21) and (7.25) that
P m k < X m ‡ k †ˆ P k < U k†
ˆ F U k† F U k†ˆ 2F U k† 1: 7:26†

We note that the result in Example 7.4 is independent of m and and is a
function only of k. Thus, the probability that X takes values within k standard
deviations about its expected value depends only on k and is given by Equation
(7.26). It is seen from Table A.3 that 68.3%, 95.5%, and 99.7% of the area
under a normal density function are located, respectively, in the ranges
2 , and m
m , m 3 . This is illustrated in Figures 7.7(a)–7.7(c).
For example, the chances are about 99.7% that a randomly selected
sample from a normal distribution is within the range of m 3 [Figure
7.7(c)].

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