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

938 Chapter 29 Statistics and probability distributions
Solution Note that ∫ ∞
0
e −z dz = 1 sothat f (z) = e −z issuitableas a p.d.f.
(a)P(0z2)= ∫ 2
0 e−z dz = [−e −z ] 2 0 = 0.865
(b) P(z >1) = ∫ ∞
1
e −z dz = [−e −z ] ∞ 1
= 0.368
(c) P(z < 0.5) = [−e −z ] 0.5
0
= 0.393
EXERCISES29.4
1 f(x) =kx 2 ,k constant, −1 x1. f(x)isap.d.f.
(a) What is the value ofk?
(b) Calculate the probability thatx > 0.5.
(c) IfP(x >c) = 0.6 then what isthe value ofc?
2 f (x)is ap.d.f. forthe random variablex,which can
varyfrom0to10.ItisillustratedinFigure29.5.What
is the probability thatxlies in [2,4]?
f(x)
0 10 x
Figure29.5
Probability densityfunctionforQuestion 2.
3 Ap.d.f. isgiven by
f(z)=2e −2z
0z<∞
(a) If200 measurements ofzare made, howmany,
onaverage, willbe greaterthan1?
(b) If50% ofmeasurementsare less thank,findk.
4 Ap.d.f.,h(x),is defined by
Calculate
h(x) = 3 4 (1−x2 )
(a) P(0 x 0.5)
(b) P(−0.3 x0.7)
(c) P(|x| < 0.5)
(d) P(x > 0.5)
(e) P(x 0.7)
5 (a) Verify that
f(t)=λe −λt t 0
issuitable asap.d.f.
(b) CalculateP(t 2) if λ = 3.
−1x1
Solutions
1 (a) 1.5 (b) 0.4375 (c) −0.5848
2 0.28
3 (a) 27 (b) 0.3466
4 (a) 0.3438 (b) 0.6575 (c) 0.6875
(d) 0.1563 (e) 0.9393
5 (b) 2.479 ×10 −3
29.5 MEANVALUE
If {x 1
,x 2
,x 3
,...,x n
} is a set ofnnumbers, then the mean value of these numbers, denotedbyx,is
x =
sum ofthe numbers
n
=
∑
xi
x issometimes called thearithmetic mean.
n