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

934 Chapter 29 Statistics and probability distributions
distributedacrossvariousrangesofvaluesgivesrisetotheideaofaprobabilitydistribution.Thestudyofprobabilitydistributionsformsthemajorpartofthischapter.Themost
important distributions, the binomial, the Poisson, the uniform, the exponential and the
normal,areallincluded. Thechapter concludes withastudyofreliabilityengineering.
29.2 RANDOMVARIABLES
Engineeringquantitieswhosevariationcontainsanelementofchancearecalledrandom
variables. Some examples are listedbelow:
(1) the diameter of a motor shaft of nominal size0.2 m;
(2) the weight of a steelbox used tocontain anelectronic circuitboard;
(3) thenumberofcomponentspassingapointonafactoryproductionlinein1minute;
(4) the nominal resistance value of resistors;
(5) the length oftime a machine works withoutfailing.
All of these quantities vary. In (1), (2) and (5) the quantities vary continuously; that
is, they can assume any value in some range. For example, a motor shaft may have any
diameterbetween0.197mand0.203m;asteelboxcouldhaveanyweightbetween,say,
0.345 kg and 0.352 kg. These are examples of continuous variables. The value itself
will only be recorded to a certain accuracy, which depends on the measuring device
and the use to which the data will be put. For example, the shaft diameter may only
be measured to the nearest tenth of a millimetre. Although such a measurement is only
integerinmultiplesofatenthofamillimetre,thevariablebeingmeasurediscontinuous.
In(3)and(4)thevariablescanassumeonlyalimitednumberofvalues.Thenumberof
componentspassingapointinaminutewillbeanon-negativeinteger0,1,2,3....The
nominal resistance value of resistors has a limited number of values which is specified
by manufacturers in their catalogues. Variables such as these, which can assume only a
limitedset of values, arecalleddiscrete variables.
EXERCISES29.2
1 Isthe length oftime amachine workswithout failing
a continuous oradiscrete variable?
2 State whether the following variables are continuous
ordiscrete:
(a)the length ofabridge
(b)the number ofelectrical sockets in ahouse
(c)the length ofcable used to wire ahouse
(d)the weight ofsolderused to build acircuit board
3 Statewhether the following variablesare discreteor
continuous:
(a) the force requiredto stretch a spring bya
specifiedlength
(b) the output voltage ofasystem
(c) the height ofacolumn ofliquid
(d) the number ofresistorsin acircuit
(e) the number ofbitsofmemory ofacomputer
Solutions
1 continuous
2 (a) continuous (b)discrete
(c) continuous (d)continuous
3 (a)continuous (b)continuous (c)continuous
(d)discrete (e)discrete