[1 mark]
ii. Find the median of
X
.
[2 marks]
iii. Deduce the value of
c
given that
E
(
X
+
c
) = 3
E
(
X

c
).
[1 mark]
iv. Two independent observations of
X
are taken. Find the probability that one is
less than
1
2
and the other is greater than 2.
[2 marks]
v. Find the cumulative distribution function (cdf) of
X
.
[2 marks]
1
(b) The time,
T
years, before a machine in a factory breaks down follows the probability
density function given by
f
(
t
) =
braceleftBigg
ate

bt
t
>
0
0
otherwise
where
a
and
b
are positive constants. It may be assumed that, if
n
is a positive number,
integraldisplay
∞
0
t
n
e

bt
dt
=
n
!
b
n
+1
.
The mean of
T
is known to be 1.5. Find the values of
a
and
b
.
[2 marks]
Note:
n
! is read ”n factorial”. For example, 5! is 120 because 5! = 5 X 4 X 3 X 2 X
1.
n
! is just shorthand for
n
X (
n

1) X (
n

2) X
....
3 X 2 X 1. You should be able
to find this function in your scientific calculator.
3. The distance by road from UBC Vancouver to UBC Okanagan is 400 km. When I travel
by car, my average speed is
V
km/hr, where
V
is uniformly distributed over the interval
60
≤
V
≤
80.
(a) Write down the cumulative distribution function of
V
.
[2 marks]
(b) Obtain the cumulative distribution function of
T
, the time in hours for the journey from
UBC Vancouver to UBC Okanagan. Hence, obtain the probability density function of
T
.
[5 marks]
(c) Calculate
E
(
T
) and
var
(
T
).
[2 marks]
(d) I have to attend a meeting at UBC Okanagan which starts at 2 pm. Find the latest
time I can leave UBC Vancouver in order that the probability of my arriving in time
for the meeting should be at least 80%.
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 Summer '11
 YewWei
 Statistics, Normal Distribution, Probability theory, partner, probability density function, Cumulative distribution function