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

29.12 The uniform distribution 961
5 (a) 0.1075 (b) 0.1188
6 (a) Binomial0.0730; Poisson 0.0743
(b) 0.1923,0.1931
(c) 0.2521, 0.2510
(d) 0.2191, 0.2176
(e) 0.2634, 0.2640
29.12 THEUNIFORMDISTRIBUTION
Wenowconsideracontinuousdistribution--theuniformdistribution.Supposetheprobability
of an event occurring remains constant across a given time interval. The p.d.f.,
f (t),of such a distributiontakes the form shown inFigure 29.8.
f(t)
1 –
T
0
Area = 1
T
t
Figure29.8
Theuniform p.d.f.
The area under f (t) must equal 1 and so if the interval is of lengthT, the height of
the rectangle is 1 T .
Thep.d.f. forthe uniformdistributionisgiven by
⎧
⎨ 1
0<t<T
f(t)= T
⎩
0 otherwise
EXERCISES29.12
The probability an event occurs in an interval [a,b] is ∫ b
a f(t)dt.If0abT
thisprobabilityissimply b −a
T .WeshallmakeuseofthisdistributioninSection29.15
when we deal with reliabilityengineering.
1 Arandom variable,x,has a uniform p.d.f.with
T = 10.Calculate the probability that
(a) 1x3 (b) 1.6x9.3
(c) x 2.9 (d) x < 7.2
(e) −1<x<2
(f) 9.1<x<12.3
2 A randomvariablet has a uniform p.d.f.with
T = 1.5. Calculate the probability that
(a) 0.7t1.3 (b) 1<t<2
(c) |t| <0.5 (d) |t| >1
Solutions
1 (a) 0.2 (b) 0.77 (c) 0.71 (d) 0.72
(e) 0.2 (f) 0.09
2 (a) 0.4 (b) 0.3333 (c) 0.3333
(d) 0.3333