Ans. (a)
x
1
/
2
for 0
≤
x
≤
1, (b) 1

1
/
√
3, (c) 1/6, (d) 1
/
4.
12. Suppose
P
(
X
=
x
) =
x/
21 for
x
= 1
,
2
,
3
,
4
,
5
,
6. Find all the medians of
this distribution.
Ans. 5 is the only median
13. Suppose
X
has a Poisson distribution with
λ
= ln 2. Find all the medians
of
X
.
Ans. any number in [0
,
1].
14. Suppose
X
has a geometric distribution with success probability 1/4, i.e.,
P
(
X
=
k
) = (3
/
4)
k

1
(1
/
4). Find all the medians of
X
.
Ans.
P
(
X
≥
k
) = (3
/
4)
k
= 1
/
2 when
k
= log(1
/
2)
/
log(3
/
4) = 2
.
409. The only
median is 2 since
P
(
X
≥
2)
≥
1
/
2 and
P
(
X >
2)
<
1
/
2 implies
P
(
X
≤
2)
≥
1
/
2.
15. Suppose
X
has density function 3
x

4
for
x
≥
1. (a) Find a function
g
so
that
g
(
X
) is uniform on (0
,
1). (b) Find a function
h
so that if
U
is uniform on
(0
,
1),
h
(
U
) has density function 3
x

4
for
x
≥
1.
Ans. (a) 1

x

3
(b) (1

u
)

1
/
3
.
16. Suppose
X
1
, . . . , X
n
are independent and have distribution function
F
(
x
).
Find the distribution functions of (a)
Y
= max
{
X
1
, . . . , X
n
}
and (b)
Z
=
min
{
X
1
, . . . , X
n
}
Ans. (a)
F
(
x
)
n
, (b) 1

(1

F
(
x
))
n
17. Suppose
X
1
, . . . , X
n
are independent exponential(
λ
). Show that
min
{
X
1
, . . . , X
n
}
= exponential(
nλ
)
Functions of random variables
18. Suppose
X
has density function
f
(
x
) for
a
≤
x
≤
b
and
Y
=
cX
+
d
where
c >
0. Find the density function of
Y
.
Ans.
f
((
y

d
)
/c
) when
ca
+
d
≤
y
≤
cb
+
d
, 0 otherwise
19. Show that if
X
= exponential(1) then
Y
=
X/λ
is exponential(
λ
).
20. Suppose
X
is uniform on (0
,
1). Find the density function of
Y
=
X
n
.
Ans.
y
(1
/n
)

1
/n
for 0
< y <
1, 0 otherwise
21.
Suppose
X
has density
x

2
for
x
≥
1 and
Y
=
X

2
.
Find the density
function of
Y
.
58
Ans.
y

1
/
2
/
2
22. Suppose
X
has an exponential distribution with parameter
λ
and
Y
=
X
1
/α
.
Find the density function of
Y
. This is the
Weibull distribution
.
Ans.
αλy
α

1
exp(

λy
α
).
23. Suppose
X
has an exponential distribution with parameter 1 and
Y
= ln(
X
).
Find the distribution function of
X
. This is the
double exponential distribution
.
Ans. 1

exp(

e

x
)
24. Suppose
X
is uniform on (0
, π/
2) and
Y
= sin
X
. Find the density function
of
Y
.
The answer is called the
arcsine law
because the distribution function
contains the arcsine function.
Ans.
π

1
(1

x
2
)

1
/
2
25. Suppose
X
has density function
f
(
x
) for

1
≤
x
≤
1, 0 otherwise. Find the
density function of (a)
Y
=

X

, (b)
Z
=
X
2
.
Ans. (a)
f
(
y
) +
f
(

y
) for 0
≤
y
≤
1, (b)
{
f
(
√
z
) +
f
(

√
z
)
}
/
2
√
z
for 0
≤
z
≤
1
26.
Suppose
X
has density function
x/
2 for 0
< x <
2, 0 otherwise.
Find
the density function of
Y
=
X
(2

X
) by computing
P
(
Y
≥
y
) and then
differentiating.
Ans.
f
(
y
) = (1

y
)

1
/
2
/
2
Joint distributions
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 Spring '16
 Probability, Probability theory, Dice