Practice Problems 6
1.
A gas station sells three grades of gasoline: regular unleaded, extra unleaded, and
super unleaded.
These are priced at $1.55, $1.70, and $1.85 per gallon
*, respectively.
Let X
1
, X
2
, and X
3
denote the amounts of these grades purchased (gallons) on a
particular day.
Suppose the X
i
’s are independent with
μ
1
= 1,000,
μ
2
= 500,
μ
3
= 300,
σ
1
= 100,
σ
2
= 80, and
σ
3
= 50.
If the X
i
’s are normally distributed,
what is the probability that revenue exceeds …
a)
$2,600?
b)
$3,000?
2.
Suppose that the actual weight of "10-pound" sacks of potatoes varies from sack
to sack and that the actual weight may be considered a random variable having a
normal distribution with the mean of 10.2 pounds and the standard deviation of
0.6 pounds.
Similarly, the actual weight of "3-pound" bags of apples varies from
bag to bag and that the actual weight may be considered a random variable having
a normal distribution with the mean of 3.15 pounds and the standard deviation of
0.3 pounds.
A boy-scout troop is planning a camping trip.
If the boy-scouts buy
3 "10-pound" sacks of potatoes and 4 "3-pound" bags of apples selecting them at
random, what is the probability that the total weight would exceed 42 pounds?
3.
Let
X
1
and
X
2
be independent with normal distributions
N
(
6, 1
)
and
N
(
7, 1
),
respectively.
Find
P
(
X
1
> X
2
).
Hint:
Write
P
(
X
1
> X
2
) = P
(
X
1
– X
2
> 0
)
and determine the distribution of
X
1
– X
2
.
4.
Compute
P
(
X
1
+ 2 X
2
– 2 X
3
> 7
),
if
X
1
,
X
2
,
X
3
are
iid
with common
distribution
N
(
1, 4
).
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*
This problem was written long time ago.