Problems and results for the twelfth week
Mathematics A3 for Civil Engineering students
1. Which of the following functions can be a distribution function?
(a)
F
(
x
) =
braceleftBigg
1 + e
1
−
x
, if
x >
−
1
,
0
, otherwise
(b)
F
(
x
) =
2
−
2
x
+ 1
, if
x
≥
0
,
0
, otherwise
(c)
F
(
x
) =
braceleftBigg
1
−
e
−
x
, if
x
≥
0
,
0
, otherwise
(d)
F
(
x
) =
0
, if
x
≤
0
,
x
4
·
(4
−
x
)
, if 0
< x
≤
2
,
1
, if
x >
2
2. Which of the following functions can be a probability density function?
(a)
f
(
x
) =
2
x
, if
x >
1
,
0
, otherwise
(b)
f
(
x
) =
sin(
x
)
2
, if 0
< x <
2
,
0
, otherwise
(c)
f
(
x
) =
3
x
−
1
ln(3)
, if
x
≤
0
,
1
3
sin
parenleftBig
x
2
parenrightBig
, if 0
< x < π,
0
, otherwise
(d)
f
(
x
) =
braceleftBigg
2e
−
2
x
, if
x >
0
,
0
, otherwise
3. Compute the expectation and variance of a random variable
X
with density
f
(
x
) =
braceleftBigg
2
x
, if 0
< x <
1
,
0
, otherwise.
4. Find the probabilities
P
{
m
−
σ < X < m
+
σ
}
and
P
{
m
−
2
σ < X < m
+2
σ
}
in the previous
question, where
m
stands for expectation and
σ
2
for variance.
5. Consider the function
f
(
x
) =
c
(2
x
−
x
3
)
, if 0
< x <
5
2
,
0
, otherwise.
Could
f
be a probability density function? If so, determine
c
.
1
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Repeat if
f
(
x
) were given by
f
(
x
) =
c
(2
x
−
x
2
)
, if 0
< x <
5
2
,
0
, otherwise.
6. A filling station is supplied with gasoline once a week.
If its weekly volume of sales in
thousands of liters is a random variable with probability density function
f
(
x
) =
braceleftBigg
5(1
−
x
)
4
, if 0
< x <
1
,
0
, otherwise,
what needs the capacity of the tank be so that the probability of the supply’s being exhausted
in a given week is 0.01?
7. Compute
E
(
X
) if
X
has a density function given by
(a)
f
(
x
) =
1
4
x
e
−
x/
2
, if
x >
0
,
0
, otherwise;
(b)
f
(
x
) =
braceleftBigg
c
(1
−
x
2
)
, if
−
1
< x <
1
,
0
, otherwise;
(c)
f
(
x
) =
5
x
2
, if
x >
5
,
0
, otherwise?
8. The lifetime, in days, of a part of a machine is a random variable with density
f
(
x
) = 2
/x
3
when
x >
1. What is the probability that this part still works on February 1, if we bought
it on January 26? Should we rather buy the part with density function
f
(
x
) = 1
/x
2
when
x >
1? What is the average lifetime for the two types of parts?
9. What is the probability that, if I wake with a start in the middle of the night, the minute
hand of the clock is on the right handside of the vertical line that passes through the center
of the clock? And what is the probability that the minute hand points inside the
1
12
part of
the circle that is located between the numbers 5 and 6?
10. What is the probability that, out of three independent points chosen in (0
,
1), precisely one
falls in each of the intervals (0
,
1
3
), (
1
3
,
2
3
), (
2
3
,
1)?
11. A long and high fence consists of columns of diameter
D
, placed in a distance of
L
from each
other. I throw a ball of diameter
d
from a long distance, with my eyes closed to this fence.
The ball either hits one of the columns, or it passes between them without touching. What
is the probability that the ball passes without touching?
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 Spring '11
 dat
 Normal Distribution, Probability, Variance, Probability theory, probability density function

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