1.) Suppose that
X
is a continuous random variable with density
p
X
(
x
) =
(
Cx
(1

x
)
if
x
∈
[0
,
1]
0
if
x <
0 or
x >
1
.
(a) Find
C
so that
p
X
is a probability density function.
(b) Find the cumulative distribution of
X
.
(c) Calculate the probability that
X
∈
(0
.
1
,
0
.
9).
(d) Calculate the mean and the variance of
X
.
2.) Suppose that
X
is a continuous random variable with cumulative distribution function
F
X
(
x
) =
1
π
arctan(
x
) +
1
2
.
(a) Find the probability density function of
X
.
(b) Calculate the probability that
X
≥
1.
(c) Show that the moment generating function
m
X
(
t
) =
∞
for any
t
6
= 0.
3.)
Let
V
= 1

U
, where
U
is a standard uniform random variable.
Find the cumulative
distribution function and the probability density function of
V
and use your results to conclude
that
V
is also a standard uniform random variable.
4.) Suppose that
X
is exponentially distributed with parameter 1 and let
Y
=
γX
, where
γ >
0
is a positive constant. Find the cumulative distribution function and the density of
Y
and use
this to identify the distribution of
Y
.
5.)
Assuming that the time that it takes a radioactive nucleus to decay is exponentially dis
tributed, find the relationship between the rate of decay
λ
and the halflife
t
1
/
2
, which is the
time at which the probability of decay is equal to 1
/
2. Given that the halflife of carbon14 is
5730 years, approximately what proportion of a sample of carbon14 will remain after 44,000
years?
6.)
Suppose that
X
1
,
· · ·
, X
n
are independent exponential random variables with parameters
λ
1
,
· · ·
, λ
n
.
Find the distribution of
Y
= min
{
X
1
,
· · ·
, X
n
}
.
Hint:
Calculate the probability
P
(
Y > t
).
2