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

Review exercises 29 977
Solutions
1 0.999
2 0.9998
3 0.9967
5 0.865
6 0.960, 0.992
7 0.940, 0.985
4 0.9901
REVIEWEXERCISES29
1 Find the mean and standard deviation of
8,6.9,7.2,8.4,9.6,10.3,7.4,9.0
2 Adiscrete random variable, v,has a probability
distributiongiven by
v −2 −1.5 −1 −0.5 0 0.5 1.0
P(v) 0.23 0.17 0.06 0.03 0.17 0.31 0.03
Calculate
(a)P(v = −1) (b)P(v = −2or v =0)
(c)P(v 0) (d)P(v < −0.5)
3 Acontinuous random variable,x, has ap.d.f., f (x),
given by
f(x)=3x 2
(a) FindP(0 x0.5).
(b) FindP(x > 0.3).
(c) FindP(x < 0.6).
0x1
(d) Find the expected value ofx.
(e) Find the standard deviation ofx.
4 Adiscrete random variable,y, has aprobability
distributiongiven by
y −0.25 0.25 0.75 1.25 1.75
P(y) 0.25 0.20 0.10 0.15 0.30
(a) Find the expected value ofy.
(b) Find the standard deviation ofy.
5 Therandom variable,t,has ap.d.f.,H(t),given by
H(t)=λe −λt
t0
(a) Calculate the expected value oft.
(b) Calculate the standard deviation oft.
6 Calculate the number ofpermutations of
(a) five distinct objects,taken two atatime
(b) eight distinct objects,taken four at atime
7 Calculate the number ofcombinations of
(a) seven distinct objects,taken four at atime
(b) 200 distinct objects,taken 198 atatime
8 The probability a component is manufactured to an
acceptable standard is 0.92.Twelve components are
picked at random. Calculate the probability that
(a) sixare acceptable
(b) 10are acceptable
(c) more than10are acceptable
(d) atleast two are notacceptable
9 The number ofbreakdowns ofacomputer systemin a
month isarandom variable with aPoisson
distribution. The mean number ofbreakdowns per
month istwo. Calculate the probability that the
number ofbreakdowns ofthe system in a monthis
(a) two
(b) three
(c) one
(d) more thantwo
10 The probability ofacomputer failingin asystem in a
given week is 0.03. Inasystem thereare 150
computers.
(a) Usethe binomial distribution to calculate the
probability that in a given week fivecomputers
fail.
(b) Usethe binomial distribution to calculate the
probability that in a given week more than two
computers fail.
(c) Usethe Poissonapproximation to calculate the
probability that in a given week four computers
fail.