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Lecture 17 on BST 631: Statistical Theory I – Kui Zhang, 10/18/2007 Review for the previous lecture Definition:
n-dimensional random vector, joint pmf (pdf), marginal pmf (pdf)
Theorem:
How to calculate marginal pmf (pdf) given joint pmf (pdf)
Example:
How to calculate marginal pmf (pdf), joint pmf (pdf)
Example:
How to calculate probability based on marginal pmf (pdf), joint pmf (pdf)
Chapter 4 – Multiple Random Variables
Chapter 4.2 – Conditional Distributions and Independence Definition 4.2.1:
Let
)^ be a discrete bivariate random vector with joint pmf
(^
, X Y^
( , f x y
and marginal pmfs
f^ xX
and
(^ ) f y^ Y
. For any such
x^
such that
)^
X
P X
x^
f^
x
=^
=^
>^
, the
conditional pmf
of^
Y^ given that
X^
x = is
the function of
denoted by
y^
|^ )
f^ y
x^
:^
(^ |
( X
)^
(^
|^
f^ y
x^
P Y
y^ X
x^
f^ x y
f
x
=^
=^
=^
X^
Y
. For any such
y^
such that
(^
)^
P Y
y^
f =^
=^
>^
, the
conditional pmf
of^
given that
y
(^ ) y^ Y
=^
is the function of
x^
denoted by
|^
f^ x
y^
)^
(^ )
(^ |
)^
(^
Y
f^ x
y^
P =
= X x
Y^
y^
f^ x y
f^
y
=^
=^
Note:
(^ | f y^
x^
is a valid pmf since
(^ |
)^
f^ y
x^
≥^
, because
)^
f^ x y
)^
and
being joint and marginal pmfs,
and since
∑^
f^ xX
(^ |
)^
)^
( X
y^ x
f^ x y
f^
x
=^
/^
( ) X
X
f^
x^
f^
x
y^
y
f
1
Lecture 17 on BST 631: Statistical Theory I – Kui Zhang, 10/18/2007 Example 4.2.
:^ Define the joint pmf of
by (^
,^ ) X Y^
f^
f =
=^
,^ (1,10)
f^
f =
=^
f^
=^
,^
f^
=^
, that is , y x^
(^ ) f yY
(^ ) f x^ X
(a) Obtain the conditional distribution of
given Y
X^
x =.
(b) Find
(^
P Y
X
>^
Solution: Definition 4.2.3:
Let
be a continuous bivariate random vector with joint pdf (^
,^ ) X Y^
f^ x y
and marginal pdfs
( ) f x^ X
and
(^ ) f y^ Y
. For any such
x^
such that
0 , the
conditional pdf
of^
Y^ given that
X^
x
f^ xX
=^
is the function of
denoted by
|^ )
y^
( f y^
x^ :
|^
)^
( ,^
) /^
X
f^ y
x^
f^ x y
f^
x
=^
. For any
y^
such that
(^ )
, the
conditional pdf
of^
X^
given
that
Y^
y^ is the function of
f^ y^ Y
=^
x^ denoted by
|^
f^ x
y^
:^
(^ |
( ,^
)^
(^ ) Y
f^ x
y^
f^ x y
f^
y
=^
Example 4.2.4:
Let the continuous random vector
have joint pdf (^
,^ ) X Y
,^
{(^ , ):
( ,^
)^
y
X Y^
u v^
u^ v^
}^
f^
x y
e^
I^
x y
−
< < <∞
=^
(a) Find the marginal pdf of
. X
(b) For any
x^
such that
, find ( )
f^ xX
>^
(^ |
f^ y
x^
2
Lecture 17 on BST 631: Statistical Theory I – Kui Zhang, 10/18/2007 Definition 4.2.5:
Let
be a bivariate random vector with joint pdf or pmf (^
,^ ) X Y^
f^ x y
and marginal pdfs or pmfs
f^ xX
and
(^ ) f y^ Y
. Then
and
Y^
are called
independent random variables
, if for every
X^
x^ ∈
R^
and
y^ ∈
R^ ,
( ,^
)^
(^ )
( ) X
Y
f^ x y
f^
x f
y
=^
.^ Consequently, if
X^
and
Y^
are independent,
(^
|^ )
( Y
f^ y
x^
f^
y =^
and
(^ |
)^
( X
f^ x
y^
f^
x =^
Technical Note:
If^
f^ x y
is the joint pdf for the continuous random vector
where (^
,^ ) X Y
( ,^
)^
X^
Y
f^ x y
f^
x f
y
≠^
on a set
A^
such that
)^
f^ A x y dxdy
∫∫^
, then
and
are still called independent random
variables.
X^
Y
Example 4.2.6:
Consider the discrete bivariate random vector
, with joint pmf given by (^
,^ ) X Y
f^
f^
f
=^
=^
=^
,^ (10,2)
f^
f =
=^
, and
f^
=^
Find the marginals of
and
. Are
and
independent?
X^
Y^
X^
Y
Solution: Question:
Can we check for independence without knowing the marginals?
Lemma 4.2.7:
Let
)^ be a bivariate random vector with joint pdf or pmf
(^
, X Y^
( , f x yy
. Then
and
are
independent random variables if and only if there exist functions
g x
and
h^
such that, for every
X^
Y
(^
(^
x^ ∈
R^
and
y^ ∈
R^ ,
( ,^
)^
f^ x y
g x h y =
. In this case
f^ xX
cg x =
and
(^ ) Y
f^
y^
dh^
y =^
, where c and d are some constants that
would make
f^ xX
and
(^ ) f y^ Y
valid pdfs or pmfs.
4
Lecture 17 on BST 631: Statistical Theory I – Kui Zhang, 10/18/2007 Example 4.2.8:
Consider the joint pdf
2 4
(^ / 2
)^ ,
)^
y^ x
f^ x y
x y e
−^ −
=^
x^ >
and
. By Lemma 4.2.7,
and
are
independent random variables.
y^ >
X^
Y
Notes:
1.^
Consider the set {( ,
and y
x^ y
x
A
B
∈^
∈^
, where
{^ :
X
A^
x^
f^
x
=^
>^
and
{^
:^
Y
B^
y^
f^
x
=^
>^
, then this set is
called a cross-product denoted by
A^
B ×.
2.^
If^
f^ x y
is a joint pdf or pmf such that the set
)^
x^ y
f^ x y
is not a cross-product, then the random
variables
X^
and
Y^
with joint pdf or pmf
f^ x y
are not independent.
3.^
If it is known that
X^
and
Y^ are independent random variables with marginal pdfs (or pmfs)
( ) f x^ X
and
(^ ) f y^ Y
, then the joint pdf (or pmf) of
X^
and
Y^
is given by
)^
X^
Y
f^ x y
f^
x f
y
=^
. (See Example 4.2.9 for
discrete case.) Theorem 4.2.10:
Let
and
be independent random variables.
X^
Y
a. For any
A^
⊂^ R
and
B^
⊂^ R
,^
(^
,^
)^
(^
)^ (
P X
A Y
B^
P X
A P Y
B
∈^
∈^
=^
∈^
b. Let
)^ be a function only of ( g x
x^ and
)^ be a function only of
. Then
( h y^
y^
(^ (
(^ (
E^ g X h Y
E g X
E h Y
=^
Proof: Example 4.2.11:
Let
and
be independent exponential(1) random variables.
X^
Y
a. Find the joint pdf of
and
X^
Y
b. Find
(^
4,^
P X
Y
≥^
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