1.) Suppose that
X
and
Y
are jointly continuous random variables with joint density function
f
(
x, y
) =
c
(
y
2

x
2
)
e

2
y
,

y
≤
x
≤
y,
0
< y <
∞
.
(a) Find
c
so that
f
is a density function.
(b) Find the marginal densities of
X
and
Y
.
(c) Find the expected value of
X
.
2.) Suppose that
X
and
Y
are jointly continuous random variables with joint density function
f
(
x, y
) =
xe

x
(
y
+1)
,
0
≤
x, y <
∞
.
(a) Find the conditional densities of
X
and
Y
.
(b) Find the cumulative distribution function and density of
Z
=
XY
.
3.)
Suppose that
X
1
,
· · ·
, X
n
are independent normal random variables with distributions
N
(
μ
1
, σ
2
1
)
,
· · ·
,
N
(
μ
n
, σ
2
n
), respectively, and let
X
=
c
1
X
1
+
· · ·
+
c
n
X
n
, where
c
1
,
· · ·
, c
n
are
real numbers. Find the distribution of
X
.
4.) Let
Z
1
, Z
2
be independent exponential random variables each with parameter 1. Find the
cumulative distribution function of the random variable
X
=
Z
1
Z
1
+
Z
2
and identify the distribution of
X
.
5.)
Suppose that
X
1
,
· · ·
, X
n
are independent, identicallydistributed random variables with
marginal density function
f
(
x
). Calculate the probability
P
(
X
1
< X
2
<
· · ·
< X
n
)
.
6.) For each
n
≥
1, let
X
n
be a Poissondistributed random variable with parameter
λ
n
=
n
and define
Y
n
=
n

1
/
2
(
X
n

n
)
.
Show that the sequence (
Y
n
;
n
≥
1) converges in distribution to a standard normal random
variable.
2