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

936 Chapter 29 Statistics and probability distributions
29.4 PROBABILITYDENSITYFUNCTIONS--CONTINUOUS
VARIABLE
Suppose x is a continuous random variable which can take any value on [0, 1]. It is
impossible to list all possible values because of the continuous nature of the variable.
There are infinitely many values on [0, 1] so the probability of any one particular value
occurring is zero. It is meaningful, however, to ask ‘What is the probability ofxfalling
inasub-interval,[a,b]?’Dividing[0,1]intosub-intervalsandattachingprobabilitiesto
each sub-interval will result in a probability distribution. Table 29.2 gives an example.
The probability thatxwill lie between 0.4 and 0.6 is 0.35, that isP(0.4 x < 0.6) =
0.35. Similarly,
P(0.2 x < 0.4) = 0.25
Figure 29.2 shows the table in a graphical form. By making the sub-intervals smaller a
more refined distributionisobtained. Table 29.3 and Figure 29.3 illustratethis.
The probability thatxlies in a particular interval is given by the sum of the heights
of the rectangles on that interval. For example, the probability thatxlies in [0.5, 0.8] is
0.2+0.1+0.1 = 0.4;thatis,thereisaprobabilityof0.4thatxliessomewherebetween
0.5 and 0.8.Note thatthe sum ofall the heights is1,representing total probability.
Considerthesub-interval[a,b].Werequiretheprobabilitythatxliesinthisinterval.
The way this is answered is by means of a probability density function (p.d.f.), f (x).
Suchap.d.f.isshowninFigure29.4,whereitistheareaunderthegraphbetweenaand
Table29.2
Probability thatxlies in agiven sub-interval.
x [0,0.2) [0.2, 0.4) [0.4,0.6) [0.6,0.8) [0.8,1.0]
P(x) 0.1 0.25 0.35 0.2 0.1
Table29.3
Refining the sub-intervals in Table 29.2.
x [0,0.1) [0.1, 0.2) [0.2,0.3) [0.3,0.4) [0.4, 0.5)
P(x) 0.03 0.07 0.1 0.15 0.15
x [0.5, 0.6) [0.6, 0.7) [0.7,0.8) [0.8,0.9) [0.9, 1.0]
P(x) 0.2 0.1 0.1 0.07 0.03
P(x)
0.4
0.3
0.2
P(x)
0.2
0.1
0.1
0
0.2 0.4 0.6 0.8 1.0
x
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
x
Figure29.2
Plotted data ofTable 29.2.
Figure29.3
Plotted data ofTable 29.3.