Statistics 103: Lab 4
LAB PROBLEMS
1. Book problem 8.3 (This one is a bit tricky. You can
assume the two students are chosen with replace
ment.
Hint: use the Binomial PMF to figure out
the marginals; figure out which entries of the joint
PMF are zero; then fill in the rest).
2. Book problem 8.10.
3. Book problem 8.14.
4. Suppose
X
and
Y
are independent random vari
ables with
E
(
X
) =
μ
1
,
var
(
X
) =
σ
2
1
,
E
(
Y
) =
μ
2
,
and
var
(
Y
) =
σ
2
2
.
Compute
E
(2
X

0
.
1
Y
+ 5),
E
(
μ
1
X
+
σ
2
2
Y
+
μ
2
),
var
(

3
Y
+
μ
2
X
+ 999),
SD
(5
X

14
Y
).
answer:
2
μ
1

0
.
1
μ
2
+5
;
μ
2
1
+
σ
2
2
μ
2
+
μ
2
;
9
σ
2
2
+
μ
2
2
σ
2
1
;
p
25
σ
2
1
+ 196
σ
2
2
PRACTICE PROBLEMS
1. Let us assume that
X
1
, X
2
, X
3
, . . . , X
n
are indepen
dent normal random variables mean
μ
and variance
σ
2
.
That is,
X
i
∼
N
(
μ, σ
2
) for
i
= 1
,
2
,
3
, . . . , n
.
Let
Y
=
∑
n
i
=1
X
i
.
Find
EY
,
E
(
¯
X
),
E
(
Y

n
¯
X
),
var
(
Y
),
var
(
¯
X
),
var
(
Y

n
¯
X
),
var
(
μ
+
σ
2
), Distribution of
Y
, Dis
tribution of
¯
X
. For
var
(
Y

n
¯
X
), simplify
Y

n
¯
X
first before solving the question.
answer:
nμ
;
μ
;
0
;
nσ
2
;
σ
2
n
,
0
,
0
,
N
(
nμ, nσ
2
)
,
N
(
μ,
σ
2
n
)
2. Suppose that you have a box which contains cards.
Each card has either “0” or “1” is written on it.
Suppose further that when drawing a card at ran
dom, the probability that you see “1” on it is
p
. Now randomly draw
n
cards with replacement
and let
X
1
, X
2
, . . . , X
n
denote the
n
card numbers.
Then, we say that
X
i
has a Binomial distribution
based on 1 trial with probability of success
p
, writ
ten
X
i
∼
Bin
(1
, p
). Now, consider
Y
=
∑
n
i
=1
X
i
.
Then,
Y
is the total of the numbers written on
n
cards. It is known that
Y
has a Binomial distribu
tion with parameters (
n, p
), or
Y
∼
Bin
(
n, p
).