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

29.4 Probability density functions -- continuous variable 937
f(x)
0 a b
x
Figure29.4
Shaded area representsP(a xb).
bthatgivesP(a xb).
P(a xb) = areaabove[a,b] =
The total area under a p.d.f. isalways 1.
∫ b
a
f(x)dx
Example29.1 Supposexisacontinuousrandomvariabletakinganyvalueon[1,4].Itsp.d.f., f (x),is
given by
f(x)= 1
2 √ x
1x4
(a) Check that f (x)isasuitable function forap.d.f.
(b) What isthe probability that (i)xlieson [2, 3.5],(ii)x 2,(iii)x < 3?
Solution (a) x can have any value on [1, 4]. For f (x) to be a p.d.f., then the total area under it
should equal 1,thatis
∫ 4
1
∫ 4
1
f(x)dx=1
f(x)dx=
∫ 4
1
1
2 √ x dx=[√ x] 4 1 = 1
Hence f (x) isasuitablefunction forap.d.f.
(b) (i) P(2 x 3.5) =
(ii) P(x 2) =
(iii) P(x < 3) =
∫ 4
2
∫ 3
1
∫ 3.5
2
f(x)dx=[ √ x] 3.5
2
= 0.457
f(x)dx=[ √ x] 4 2 = 0.586
f(x)dx=[ √ x] 3 1 = 0.732
Example29.2 Arandomvariable,z, has a p.d.f. f (z)where
f(z)=e −z
0z<∞
Calculate the probabilitythat
(a) 0z2
(b) z ismorethan1
(c) z islessthan0.5