• Name:
Score:
Math 5080, Statistical Inference I, Fall 2015
Midterm 1
Instructions
1. Write all solutions in the space provided, and use the back pages if you have to.
2. For full credit you must show all work. Providing only an answer will result in very few marks.
3. Calculators are allowed but won’t help much.
4. If you don’t understand the wording of any of the questions feel free to ask.
5. One page of notes is allowed.
1. (20 points total)
Let X and Y have the joint pdf f(x, y)
(a) Find the joint pdf of U
32 for 0
6e
X/Y and V
x
<
<
and 0
<
y
<
oo, and zero otherwise.
X.
(b) Find the marginal pdf of U.
V/
y
I
•P
(u
I
y)
1
(X
0
V/L
hAL
-3’_2(v7)
o ‘
j
4
(
/
—
7..
—6
-
—
u_
(3
‘)
)
-(3k
2.
/
I
—
(/)
{e(32]
Le
p(
.
2. (20 points total)
Let X
,X
1
,
2
.
.
,
6 be iid from the distribution with pdf
X
I
(x+1)/2,
0,
—1<x<1
otherwise
Find the probability that exactly four of the observations are larger than zero.
(
sJ
.ç
X
)o)
II
()
(y)i
9
()
L
1
-,
X1L
S
y
V
II
—
C
-
—
I
N
0
0
—
x
I
x
—
-
x
I
.•
-••
I,
0
x
‘I
—\
I
(I.
I
x
0
?c
\j
T.
r-4
0
L
-4-
(
—
-
>(,
•
—
‘I
(
-
I’
x
>::
x
—
0
4. (20 points total)
Let 1
XX
,
,
2
5 be iid from the exponential(1) distribution,
X
0 < x < co. Let Y = X() be the order statistics.
.
.
.
,
meaning
that their pdfs are
f(x)
=
e for
2 and Y
(a) Find the joint pdf of Y
.
1
(b) Show that U
1
2 and U
Y
2
-
are independent.
—
e
(
3
1
[i
‘
-yi
Je
-
F
Le e
Et F&i
-
[ -3
-
_
1
_
-Z2y
OyyoO
U
U
o
(, u)
51
5 i
-u
•-
\
•1
-
—U
—e
u
2
_
07
-c
UI
5. (20 points total)
Let X
,X
1
,
2
.
.
,
5 be iid N(0, u
X
) random variables.
2
(a) Find the constant c so that
1 -x
x
2
C
+
+
has a Student’s t—distribution.
(b) How many degrees of freedom are associated with this T random variable?
N(
e
—
6
T7No
-
3
-t
1
(3)
-‘
)t’(i’)
X-)
t)K)/
1
v3
Extra Credit (2 Points Each)
1. State the strong law of large numbers. including assumptions.
I,
E C Ix1]
.,
fC
0,?
2. State the Central Limit Theorem,
2<
including assumptions.
—[‘1
r
-
c o/e.y
,X
1
,
2
3. If X
.
,
•-.
cJ;4.;
X are iid and N([z, u
) distributed, what is the distribution of the sample mean
2
.-%/
I
PA—’