BES Tutorial Sample Solutions, S1/11
WEEK 5 TUTORIAL EXERCISES (To be discussed in the week starting
March 28)
1.
The manager of a factory has determined from past experience that
X
, the
number of repairs required to machines in her factory on any one day, has
the following probability distribution:
x
0
1
2
3
4
P(
X = x
)
0.41
0.25
0.18
0.10
0.06
Calculate the following probabilities:
TN: (c) & (d) are not probabilities. Above is a typo.
(a)
P(1 <
X
< 4)
P(1 < X < 4) = P(X=2)+P(X=3) = 0.18+0.10 = 0.28
(b)
P(0
≤
X
≤
3)
P(0
≤
X
≤
3) = P(X=0)+P(X=1)+P(X=2)+P(X=3) = 1P(X=4) = 0.94
(c)
ܧሺܺሻ ൌ ߤ ൌ ݔܲሺܺ ൌ ݔሻ
௫
ൌ 0 ൈ 0.41 1 ൈ 0.25 2 ൈ 0.18 3 ൈ 0.1 4 ൈ 0.06 ൌ 1.15
(d)
ܸܽݎሺܺሻ ൌ ܧሺܺ െ ߤሻ
ଶ
ൌ ߪ
ଶ
ൌ ሺݔ െ ߤሻ
ଶ
ܲሺܺ ൌ ݔሻ
௫
ൌ ሺ0 െ 1.15ሻ
ଶ
ൈ 0.41 ሺ1 െ 1.15ሻ
ଶ
ൈ 0.25 ሺ2 െ 1.15ሻ
ଶ
ൈ 0.18
ሺ3 െ 1.15ሻ
ଶ
ൈ 0.10 ሺ4 െ 1.15ሻ
ଶ
ൈ 0.06 ൌ 1.5075
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2.
Suppose that the daily number of errors a randomlyselected teller makes is
denoted by
X
and follows the distribution given in the table below. A
human resource manager records the daily numbers of errors of two
randomly selected tellers. Denote the associated random variables by
X
1
and
X
2
. As the selection is random,
X
1
and
X
2
are independent and follow
the same distribution as
X
. The manager then computes the sample mean
ܺ
ത
ൌ
భ
ା
మ
ଶ
where the sample size is
n
= 2.
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 Three '11
 DenzilGFiebig
 Variance, Probability distribution, Probability theory, probability density function

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