Statistics 103: Lab 5
LAB PROBLEMS
1. Suppose
X
and
Y
are the random variables with
the joint PMF:
X
\
Y
1
2
3
1
0.1 0.3 0.1
2
0.2
0 0.1
3
0 0.1 0.1
(a) Compute
cov
(
X, Y
).
(b) Compute
var
(
X
)
(c) Compute
var
(
Y
)
(d) Compute
var
(
X
+
Y
) using the variance for
mula.
answer:
0
.
1
;
0
.
61
;
0
.
6
,
1
.
41
2. Suppose that
Z
0
, Z
1
, Z
2
, . . .
are a sequence of in
dependent normal random variables with mean 0
and variance
σ
2
. Let a stock price on day
t
be de
scribed by
X
t
=
a
+
bZ
t
+
cZ
t

1
, t
= 1
,
2
,
3
, . . .
,
where
a, b
, and
c
are all positive constants.
Find
Var(
X
t
), Cov(
X
t
, X
t
+1
), and Cov(
X
t
, X
t
+2
). From
your calculations, does knowing
X
t
give you insight
into what
X
t
+1
or
X
t
+2
might be?
answer: Var
(
X
t
)
=
(
b
2
+
c
2
)
σ
2
; Cov(
X
t
, X
t
+1
) =
bcσ
2
; Cov(
X
t
, X
t
+2
) = 0; knowing
X
t
should help
us predict
X
t
+1
, but not
X
t
+2
.
3. Consider randomly picking a married couple from
Minnesota. Let
W
denoted the height of the wife
(in inches) and let
H
denoted the height of the
husband (in inches). Pearson and Lee published the
following approximations for the expected values,
standard deviations and correlation between
H
and
W
:
EW
= 63
,
sd
(
W
) = 2
.
5
(1)
EH
= 68
,
sd
(
H
) = 2
.
7
(2)
ρ
= 0
.
25
(3)
(a) If
W
= 66 what do you predict the husbands
height to be?
(b) What percentage of married women from Min
nesota are over 5 feet 8 inches tall?
(c) Of the women who are married to men 6 feet
tall, what percentage are over 5 feet 8 inches
tall?
answer: 68.81; 2.28%; 4.65%
4. Suppose
X
1
, ...X
15
is random sample (with replace
ment) from the population with mean 15 and stan
dard deviation 10. Please approximate the follow
ing probabilities using the Central Limit Theorem.
(a)
P
(
X >
9)
(b)
P
(13
.
5
<
X <
20
.
1)
answer: (a) 0.9898 , (b) 0.6952
5. Let
X
and
Z
be two independent standard normal
random variables.
Define
Y
to be
aX
+
b
+
Z
.
Compute the correlation between
Y
and
X
. If you
observe that
X
= 1
.
28 what do you predict for
Y
?
Does this prediction make sense to you?
answer:
a
√
a
2
+1
; a(1.28) + b; hopefully.
PRACTICE PROBLEMS
1. In Major League Baseball, two commonly used
metrics for the evaluation of pitching are ERA
(earned run average) and WHIP (walks plus hits
per inning pitched). A scatterplot of all 30 teams’
ERA and WHIP is shown in Figure 1. Use the scat
terplot to estimate the standard deviations of ERA
and WHIP and the correlation between ERA and
WHIP. Use these values to estimate the covariance
between ERA and WHIP. Also, predict the WHIP
of a team with an ERA of 4.5.
Now, suppose that the variances of ERA and WHIP
are 0.5 and 0.01, respectively, and the correlation
between the two metrics is 0.9. Calculate the co
variance between ERA and WHIP and compare it
to your previous estimate.
answer: 0.0636
2. After he gets out of school each day, Calvin spends
his remaining waking hours doing his homework,
watching TV, or playing Calvinball with Hobbes.
Suppose that Calvin gets out of school at 3:00pm
every day and goes to sleep at 9:00pm. Let
X
be the
number of hours that Calvin spends doing home
work on a given night, and let
Y
be the number
of hours he spends playing Calvinball. If Calvin’s
parents force him to do homework for exactly two
hours every night, what is the correlation between
X
and
Y
?