wherethe factor is calledthenormalizationconstant.It is neededto makethe probabilityover
the entirespaceequalto That is,Example7:
Supposethat boxescontaining tea bagshavea meanweightof ounceseachand a standard
deviationof ounce.
1.Whatpercentageof all the boxesis expectedto weighbetween and ounces?
2.Whatis the probabilitythat a box weighsless than ounces?
3.Whatis the probabilitythat a box will weighexactly ounces?
Solution:
- Usingthe normalprobabilitydensityfunction,
Substitutingfor and we get
The percentageof all the tea boxesthat are expectedto weightbetween and ouncescan be cal-
culatedas
The integralof doesnot havean elementaryanti-derivativeand thereforecannotbe evaluatedby
standardtechniques.However, we can use numericaltechniques,suchas The Simpson’s Ruleor The
TrapeziodRule,to find an approximate(but very accurate)value.Usingthe programingfeatureof a scientific
calculatoror, mathematicalsoftware,we eventuallyget
That is,