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ISyE 2027
Probability with Applications
Fall 2006
R. D. Foley
Homework 8
November 9, 2006
due on Tuesday
1. Suppose
X
is a continuous random variable with probability density function
f
(
s
) =
cs
2
for 0
≤
s
≤
1. Compute the following: (a)
c
, (b) Pr
{
X
≤
2
/
3
}
, (c) Pr
{
X >
2
/
3
}
, (d) Pr
{
X
= 2
/
3
}
, (e)
Pr
{
X
≤
1
/
3

X
≤
2
/
3
}
, (f) Pr
{
X
≤
2
/
3

X
≤
1
/
3
}
, Pr
{
X
≤
3
}
, and Pr
{
X
≤ 
4
}
.
2. For the previous problem, see if you can determine some function
F
(
t
) so that
F
(
t
) = Pr
{
X
≤
t
}
for
∞
< t <
∞
.
3. Suppose
Y
is a continuous random variable with p.d.f.
c

s

for

1
≤
s
≤
1. Determine
c
.
4. At the beginning of the course, we talked about the angle formed by the valve stem on the right front
time of particular student’s car. Let
U
be the angle. Assume that if the valve was at the very front
of the tire, then
U
= 0. If the valve were at the top of the tire, then
U
=
π/
2. What would be a
reasonable choice for the p.d.f. of
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