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

29.7 Expected value of a random variable 945
f(x)
Area ≈ f(x).dx
x
x + dx
x
Figure29.6
The shaded area represents the probability thatxlies
in the smallinterval [x,x + δx].
Example29.9 Arandom variable has p.d.f. given by
f(x)= 1
2 √ x
1x4
Calculate the expected value ofx.
Solution µ =
∫ 4
1
[ x
3/2
=
3
x 1
2 √ x dx = ∫ 4
] 4
1
= 7 3
1
√ x
2 dx
So, if several values ofxare measured, the mean of these values will be near to 7 3 . As
more and more values are measured the mean will get nearer and nearer to 7 3 .
EXERCISES29.7
1 Calculatethe expected value ofthe discrete random
variable,h,whose probabilitydistribution is
h 1 1.5 1.7 2.1 3.2
P(h) 0.32 0.24 0.17 0.15 0.12
2 Calculatethe expected value ofthe random variable,
x,whose probabilitydistribution is
x 2 2.5 3.0 3.5 4.0 4.5
P(x) 0.07 0.36 0.21 0.19 0.10 0.07
4 A randomvariable,x,has p.d.f. f (x) given by
f(x)= 5
4x 2
1x5
(a) Calculate the expected value ofx.
(b) Ten values ofxare measured. They are
1.9, 2.9, 2.8, 2.1, 3.2, 3.4, 2.7, 2.3, 2.8, 2.7
Calculate the mean ofthe observations and
comment on yourfindings.
5 A p.d.f.h(x)is defined by
h(x) = 3 4 (1−x2 )
−1x1
Calculate the expected value ofx.
3 Arandom variable,z,has p.d.f. f (z) = e −z ,
0 z<∞. Calculatethe expected value ofz.
6 Is the expected value ofadiscrete random variable
necessarily oneofits possiblevalues?