082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

964 Chapter 29 Statistics and probability distributions
N(x)
N(x)
N(x) (a) (b)
(a)
m
x
(b)
m
x
m 1 m 2 x
Figure29.10
Two typical normal curves.
Figure29.11
Two normalcurveswith the same
standarddeviationbutdifferentmeans.
Letxbeacontinuous random variable with a normal distribution,N(x). Then
N(x) = 1
σ √ 2π e−(x−µ)2 /2σ 2
−∞<x<∞
where µ =expected(mean)valueofx, σ =standarddeviationofx.Figure29.10shows
two typical normal curves. All normal distributions are bell shaped and symmetrical
about µ.
In Figure 29.10(a) the values ofxare grouped very closely to the mean. Such a distribution
has a low standard deviation. Conversely, in Figure 29.10(b) the values of the
variable are spread widely about the mean and so the distribution has a high standard
deviation.
Figure29.11showstwonormaldistributions.Theyhavethesamestandarddeviation
butdifferentmeans.ThemeanofthedistributioninFigure29.11(a)is µ 1
whilethemean
of that in Figure 29.11(b) is µ 2
. Note that the domain ofN(x) is (−∞,∞); that is, the
domain is all real numbers. As for all distribution curves the total area under the curve
is1.
29.14.1 Thestandardnormal
A normal distribution is determined uniquely by specifying the mean and standard deviation.
The probability thatxliesinthe interval [a,b] is
P(axb)=
∫ b
a
N(x)dx
Themathematicalformofthenormaldistributionmakesanalyticintegrationimpossible,
so all probabilities must be computed numerically. As these numerical values would
change every time the value of µ or σ was altered some standardization is required. To
this end we introduce the standard normal. The standard normal has a mean of 0 and
a standard deviation of 1.
Considertheprobabilitythattherandomvariable,x,hasavaluelessthanz.Forconvenience
wecall thisA(z).
A(z) =P(x <z) =
∫ z
−∞
N(x)dx
Figure29.12illustratesA(z).ValuesofA(z)havebeencomputednumericallyandtabulated.TheyaregiveninTable29.7.Usingthetableandthesymmetricalpropertyofthe
distribution,probabilities can becalculated.