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

978 Chapter 29 Statistics and probability distributions
(d) Usethe Poissonapproximation to calculate the
probability that in a given week more than three
computersfail.
11 Arandom variable,t,has a uniform distribution, f (t),
given by
{ 14
0<t<4
f(t)=
0 otherwise
Calculate the probability that
(a) 1t3
(b) 0.2t2.7
(c) |t| >1 (d) |t| 1.5
12 Thetime,t,betweenfailures foraparticular typeof
electrical component follows an exponential
distribution with a meanvalue of24weeks. Calculate
the probabilitythat
(a)t>24 (b)t<22
13 Theresistance ofresistorsis arandom variablewith a
normal distributionwith ameanof3ohmsanda
standard deviation of0.1ohm. Calculate the
probability thataresistor has aresistance of
(a) between 2.9and3.05 ohms
(b) more than2.95 ohms
(c) less than2.83 ohms
14 Thecapacitance ofcapacitors isarandom variable
with anormal distribution, with ameanof12farads
andastandard deviation of0.6farads.Inabatchof
300 capacitors how many would you expectto have
with capacitance
(a) between 11and12.3 farads
(b) more than11.6 farads
(c) less than12.8 farads?
15 Acomputer fails,onaverage, onceevery 4 months.
An engineering systemusesninecomputers, all of
whichneedto be workingforthe system to operate.
Calculate the probability thatthe system willfail
sometimeduring
(a) a1monthperiod
(b) a2weekperiod(i.e.half amonth)
16 TheMTBF foramotor is270 hours.When the motor
fails ittakes7hours to repair. Calculate the
availabilityofthe motor.
17 Asystem comprisesfour components in series.The
MTBF foreachcomponentis 170 hours,200 hours,
250 hours and210 hours. Calculate the reliabilityof
the system over a100 hour period.
18 Asystem comprisestwo components in parallel.The
components have anMTBF of150 hours and
170 hours. Calculate the probabilitythe systemwill
failin a 200 hourperiod.
Solutions
1 mean = 8.35;st.dev. = 1.1314
2 (a) 0.06 (b) 0.4 (c) 0.51 (d) 0.46
3 (a) 0.125 (b) 0.973 (c) 0.216
(d) 0.75 (e) 0.194
4 (a) 0.775 (b) 0.799
5 (a)
1
λ
(b)
1
λ
6 (a) 20 (b) 1680
7 (a) 35 (b) 19900
8 (a) 1.4687 ×10 −4 (b) 0.1835
(c) 0.7513 (d) 0.2487
9 (a) 0.2707 (b) 0.1804 (c) 0.2707
(d) 0.3233
10 (a) 0.1736 (b) 0.8307 (c) 0.1898
(d) 0.6577
11 (a) 0.5 (b) 0.625 (c) 0.75
(d) 0.375
12 (a) 0.3679 (b) 0.6002
13 (a) 0.5328 (b) 0.6915 (c) 0.0446
14 (a) 193 (b) 225 (c) 272
15 (a) 0.8946 (b) 0.6753
16 0.9747
17 0.1402
18 0.5093