STAT 2011 Probability and Estimation Theory - Semester 1, 2023
Tutorial Sheet Week 8
Tutorial Problems for Week 8
1. Suppose
U
is a uniform random variable over [0
,
1]. Show that
Y
= (
b
−
a
)
U
+
a
is uniform
over [
a, b
] and determine its variance.
2. Let
Y
be a uniform random variable defined over the interval (0
,
2). Find an expression
for the
r
-th moment of
Y
about the origin.
Also, use the binomial expansion to find
E
[(
Y
−
µ
)
6
].
3. Suppose that the random variable
Y
is described by the pdf
f
Y
(
y
) =
cy
−
4
,
y >
1. Find
c
and determine the highest moment of
Y
that exists.
4. Let
X
1
and
X
2
be independent random variables, both having the exponential pdf,
f
X
i
(
x
i
) =
λe
−
λx
i
,
x
i
>
0,
i
= 1
,
2. Determine
f
W
(
w
), the pdf of the sum
W
=
X
1
+
X
2
.
5. Let
f
X
(
x
) =
xe
−
x
,
x
≥
0, and
f
Y
(
y
) =
e
−
y
,
y
≥
0, where
X
and
Y
are independent. Find
the pdf of
X
+
Y
.
6. Let
Y
be a continuous non-negative random variable.
Show that
W
=
Y
2
has pdf
f
W
(
w
) =
1
2
√
w
f
Y
(
√
w
). (
Hint:
First find
F
W
(
w
).)
7. If
p
X,Y
(
x, y
) =
cxy
at the points (1
,
1), (2
,
1), (2
,
2) and (3
,
1), and equals 0 elsewhere,
find
c
.
8. An urn contains four red chips, three white chips, and two blue chips. A random sample
of size 3 is drawn without replacement. Let
X
denote the number of white chips in the
sample and
Y
the number of blue chips. Write a formula for the joint pdf of
X
and
Y
.
9. Let
X
and
Y
have the joint pdf
f
X,Y
(
x, y
) = 2
e
−
(
x
+
y
)
, 0
< x < y
, 0
< y
. Find
P
(
Y <
3
X
).
10. A man and a woman decide to meet at a certain location. If each of them independently
arrives at a time uniformly distributed between 12 noon and 1pm, find the probability
that the first to arrive has to wait longer than 10 minutes.
1