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

29.7 Expected value of a random variable 943
EXERCISES29.6
1 Calculate the means andstandard deviationsof:
(a) 1, 2, 3, 4, 5
(b) 2.1, 2.3, 2.7, 2.6
(c) 37, 26,19, 21,19,25, 17
(d) 6, 6, 6, 6, 6, 6
(e) −1, 2, −3, 4, −5, 6
2 Asetofmeasurements {x 1 ,x 2 ,x 3 , . . . ,x n }hasamean
ofxandastandard deviation ofs.What are the mean
and standard deviation ofthe set
{kx 1 ,kx 2 ,kx 3 ,...,kx n }wherek isaconstant?
3 The meanofthe numbers {x 1 ,x 2 ,x 3 , . . . ,x n }isx.
Show that the sum ofthe deviations about the mean
is 0; that is,show ∑ n
i=1 (x i −x)=0.
Solutions
1 (a) mean = 3; st.dev. = 1.414
(b) 2.425,0.238
(c) 23.4,6.321
(d) 6,0
(e) 0.5,3.862
2 mean=kx,st.dev. =ks
29.7 EXPECTEDVALUEOFARANDOMVARIABLE
InSections 29.5and29.6 weshowedhowtocalculatethe mean andstandarddeviation
ofagivensetofnumbers.Noreferencewasmadetoprobabilitydistributionsorp.d.f.s.
Supposenowthatwehaveknowledgeoftheprobabilitydistributionofadiscreterandom
variable or the p.d.f. of a continuous random variable. The mean value of the random
variable can still be found. Under these circumstances the mean value is known as the
expectedvalueor expectation.
29.7.1 Expectedvalueofadiscreterandomvariable
Supposefordefinitenessthatxisadiscreterandomvariablewithprobabilitydistribution
asgiven inTable29.5.
Table29.5
Probabilitydistribution foradiscrete randomvariablex.
x 0 1 2 3 4
P(x) 0.1 0.2 0.4 0.15 0.15
In100trialsxwillhaveavalueofzero10timesonaverage,avalueofone20timeson
average and soon. Themean value, thatisthe expected value,istherefore
expected value =
0(10) +1(20) +2(40) +3(15) +4(15)
100
We could have arranged the calculation as follows:
expected value = 0 ( ) (
10
100 +1 20
) (
100 +2 40
100
)
+3
( 15
100
= 2.05
) (
+4
15
)
100
= 0(0.1) +1(0.2) +2(0.4) +3(0.15) +4(0.15)