550.430 Introduction to Statistics
Spring 2012
Naiman
Homework #3
Due Monday Feb 20th
(1) Consider simple random sampling (without replacement) using a sample of size
n
from a population of size
N
where each individual
i
has two attributes
x
i
and
y
i
.
Let
σ
XY
=
1
N
∑
N
i
=1
(
x
i

μ
x
)(
y
i

μ
y
) denote the population covariance between
x
and
y
where
μ
x
and
μ
y
denote the population means. Let (
X
i
, Y
i
)
, i
= 1
, . . . , n
denote the
(
x, y
) values for the individuals sampled. The purpose of this exercise is to derive the
covariance between
X
i
and
Y
j
.
Assume that the values taken on by the
x
’s is
ζ
1
, . . . , ζ
u
and by the
y
’s is
ω
1
, . . . , ω
v
and that the number of (
x, y
) pairs taking the value (
ζ
i
, ω
j
) is
n
ij
.
We can think of
the population is broken into groups labeled 1
, . . . , u
according to the
x
value and into
groups labeled 1
, . . . , v
according to the
y
value. We can also think of grouping into
u
×
v
groups according to both citeria.
(a) If we draw an individual from the population with values
X
and
Y,
let
I
denote
the index telling us what
x
group this person is in (so that
X
=
ζ
I
) and let
J
denote the index telling us what
y
group this person is in. Thus (
X, Y
) = (
ζ
I
, ω
J
)
.
Write down the joint probability mass function for (
I, J
)
.
(b) Suppose we draw two individuals at random without replacement. Let (
I, J
) de
note the index pair for the first individual drawn, and let (
I
′
, J
′
) denote the index
pair for the second individual. Write down the joint probability mass function for
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 Fall '11
 Naiman
 Statistics, Probability theory, Yi, Probability mass function, sample variance, yi − µy, joint probability mass

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