HW 1
Revision of Probability
1.
X
,
Y
and
Z
are independent random variables with
E
(
X
) =
2,
E
(
Y
) =
0,
E
(
Z
) =
-
1 and
SD
(
X
) =
1,
SD
(
Y
) =
2 and
SD
(
Z
) =
3.
(a)
E
(
3
-
X
+
2
Y
+
4
Z
) =
(b)
V
(
4
-
X
+
2
Y
-
3
Z
) =
(c)
SD
(
4
-
X
+
2
Y
-
3
Z
) =
(d)
E
(
X
*
Y
*
Z
) =
(e) Calculate the
SD
(
X
*
Y
)
2. The current age (in years) of 300 clerical employees at an insurance claims processing center is normally distributed with
mean 35 and SD 5.
(a) Are most of the employees at this center are older than 25? Why or why not?
(b) Is it true that if none of these employees leave the firm and no new hires are made, then the distribution a year from
now will be normal with mean 36 and SD 6? Why or why not?
(c) Three individuals are drawn at random from this population. What are the chances that the first person is younger
than the sample average of the second and third individuals?
3. The number of packages handled by a freight carrier daily is normally distributed. On average, 1,200 packages are shipped
daily, with SD 150. Assume that the package counts are independent from one day to the next.
(a) What is the distribution of the total number of packages shipped over five days?
(b) What are the chances that the total number of packages shipped over five days is more than 6500?
(c) What is the distribution of the sample mean of the number of packages shipped over ten days?
(d) What are the chances that the sample mean of the number of packages shipped over ten days is more than 1150?
(e) What is the distribution of the difference between the numbers of packages shipped on any two consecutive days?
(f) The difference between the numbers of packages shipped today and the number of packages shipped tomorrow is
equal to zero. Would you agree with this statement?
(g) If each shipped package earns the carrier $10, then what is the distribution of the amount earned per day?
(h) What are the chances that more packages are handled tomorrow than today?
4. A taxi driver charges his passengers a $5 flat fee plus $3 per mile. Suppose that the distribution of trip distances of his
passengers is approximately normal with a mean of 10 miles and a standard deviation of 6 miles.
(a) What is the distribution of the fares?
(b) What is the probability that the fare will exceed $45?
(c) Suppose we collect information from 10 passengers (who used this taxi service) on how much money they had spent
on total. If
Y
= total amount spent by these 10 individuals, then
i.
Y
=
X
1
+
...
+
X
10
, where
X
i
s are the amount spent by the ith customer and they are i.i.d
ii.
Y
=
10
X
, where
X
= amount spent by an individual customer
iii.
Y
=
X
10
, where
X
= amount spent by an individual customer
iv.
Y
=
X
+
X
+
X
+
X
+
X
+
X
+
X
+
X
+
X
+
X
, where
X
= amount spent by an individual customer
v. None of the above
(d) What are the expected value and variance of Y?
(e) What would be the distribution of
Y
?
(f) Let
M
be the sample mean of the information collected from these 10 individuals. What is the distribution of
M
?
(g) What are the chances that out of the 10 individuals at least one of them spent more than $50.
5. If
X
is a continuous random variable, then which one of the following options are true,
P
(
b
<
X
<
a
) =
(a)
P
(
b
<
X
≤
a
)
1