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Chapter 4Chapter 4
Multiple Random Variables: Many experiments involve dealing with multiple r.v.’s and one may be interested in their interactions
or joint behavior
, e.g.,
suppose in every high school, the SAT score and the ending GPA for each student, s
, are recorded as SAT (si
) and GPA (si^
),i^
which are considered to be r.v.’s. Now, we may be interested in relating the two r.v.’s. We can also generate other r.v.’s for each student as GPA in college, etc. So far, we know how to deal with each r.v. individually but here we develop techniques to determine the
joint statistical
properties of multiple r.v.’s
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Vector Random Variables: Definition:
Consider two r.v.’s X and Y defined on a sample
space, S (universal set). If the outcome of the experiment is ξ
, then two numbers x = X(
ξ
) and y = Y(
ξ
) are created. The
ordered pair of (x, y) is called a 2-D r.v. or a “Random vector”.In a similar way, an M-D r.v. can be generated.
2
S ξ
X(
ξ) Y(
ξ)
X Y
R
: Range Space for XX
R
: Range Space for YY
x
y
(^ X(
ξ),Y(
ξ)
S ξ
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Definition-1 (Joint Cumulative Distribution Function): Probability of joint event {X
x, Y
y} is called “
joint CDF
of the r.v.’s X and Y and is denoted by,
{
}]
[
,^
y Y x X P y x F
Y X^
The domain of F
X,Y
is the set of all ordered pairs (x, y), x
R
X
and y
R
Y^
and the range of F
X,Y
is [0, 1].
{^
}^
)
(
]
,
[^
B A P y Y x X P ∩
=
≤
≤
Clearly,
as shown in the figure.
Properties of Joint CDF: Most of these follow from the definition^ (1)
(^
,
,
,^
x
F
y
F
F
Y X Y X Y X
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,^
= ∞ ∞ Y X F
,^
y x
F
Y X
,^
y x
F
Y X^
is a non-decreasing function of both x and y.
{
}]
[^
2
1
2
1
y Y y x X x P
[^
]^
[^
])
, ( ) , ( ) , ( ) ,
(^
1 1 , 1 2 , 2 1 , 2 2
,^
y x F y x F y x F y x F
Y X Y X Y X Y
X^
(^
1 2 , 2 1 , 1 1 , 2 2
,^
y x F y x F y x F y x F
Y X Y X Y X Y
X^
Marginal CDF:
,^
x
F
x
F
X
Y X^
,^
y
F
y
F
Y
Y X^
To see this, let y =
, then {Y
}^
Æ
B = S (Certain event)
and A
B = A
S = A, and
5
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Solution: (a) For x < 1, y < 1,
{
}
{^
}^
{^
}
(^1). 0
) 1 , 1 (
) , (
) 1 , 1 (
,^
,^
= = ⇒ = ≤ ≤
P y x F y Y x X
Y X
{^
}^
,^
(x, y)
F
y
Y
x
X
X,Y
{
}
{
}) (^2) , (^2) ( ), 1 , 1 (
,^
y
Y x
X
For 1
x < 2, 1
y < 2,
For 2
x < 3, 2
y < 3,
{^
}
{^
}
,^
P
P
y x
F
Y X
For x>
4, y>
{^
}^
{^
})
,^
y
Y
x
X
{^
}
{^
}
{^
}
{^
}^
,^
P P P P y x F
Y X
,
y u x u y u x u y u x u y u x u y x F
Y X
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1
2
3
4
1
2
3
4
0
FX,Y
(x,y)
x
y
1.0 There is a stepwith a corner ateach point (x
, yi
)i
(b)
{^
}
,^
Y X F Y X P
(c)
{^
}
,^
Y X
X^
F
F
X
P
Marginal Probability
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Example 2:
A joint CDF for continuous r.v.’s X and Y is,
,
xy x^ y
y x
F
Y X
and 1
and 1
and 1
and 1
or 0
y
x
y
x
y
x
y
x
y
x
(a) Sketch this joint CDF. (b) Find and sketch marginal CDF’s for both r.v.’s. Solution: (a)
[^
][
]^
[^
]
[^
]^
y u x u x u y u y u y y u
x u x u x y u y u x u x u
xy
y x
F
Y X
10
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(a)
FX,Y
(x,y)
x
y
0
1 1
xy
x y
⎧ ⎪ ⎨ ⎪⎩
∞
(^
,^
x
x
F
x
F
Y X
X
x
x
x
and
(^
,^
y
y
F
y
F
Y X
Y
(b)
y
y
y
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Properties of Joint PDF: (1)
(^
)^
,^
y x
f^
Y X
(^
)
∞ ∫ ∫∞−
,^
dxdy y x
f^
Y X^
Can be used to check forvalidity of a PDF
(^
)^
(^
)
d d f y x F
y^
x
Y X
Y X^
,^
,
,^
∫ ∫^ ∞−^
∞−
( )
(^
)^
(^
) ∞
∫ ∫∞−
∞^ ∞−
,^
,
,^
x F d d f x F Y X
x
Y X
X
(^
)^
(^
)^
(^
) y F d d f y F
Y X
y
Y X
Y^
,^
,
,^
∫ ∫∞−
∞^ ∞−
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[^
]^
(^
)^
dy dx y x f y Y y x X x P
y y
x x
Y X^
2 1
2 1
,
2
1 2
1
∫ ∫
i.e., volume under f
X,Y
(x,y).
( )
(^
)^ dy y x
f
x
f^
Y X
X^
,
∞ ∫ ∞−
(^
)^
(^
)^ dx y x
f
y f^
Y X
Y^
,
∞ ∫ ∞−
Marginal PDF
Example 3:
(a) Find
b
so that following function is a valid PDF
(^
)^
(^
)
− 0
,
y x
Y X
be
y x
f^
elsewhere
y
a
x
(b) Find their joint CDF.
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(b)
For 0
x < a and y > 0,
(^
)^
(^
)
d d f y x F
y^
x
Y X
Y X^
,^
,
,^
∫ ∫^ ∞−^
∞−
∫^
∫^
−
−
− − =
y^
x
a^
d
e
d
e
e^
0
0
α
β
(^
)(
)
(^
) a
y
x
e
e
e
−
−
−^ −
When x
a and y > 0,
(^
)^
∫^
∫^
−
−
− − =
y^
a
a
Y X^
d e d e e y x F
0
0
,^
1
1
,
α
β
(^
)(
)
(^
)^
y
a
a
y
e
e
e
e^
−
−
−
−
16
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And it is zero when x < 0 and y < 0.^ Alternately,
y
x
(^
)(
)
−
−
−
−^ y
a
y
x
Y X
e
e
e
e
y x
F
,^
y a
x
,^
y x
a
Example 4:
A fair coin is tossed twice. Let r.v. X be number
of heads (0 or 1) on the first toss and r.v. Y be number of headson the second toss. Find and sketch their joint PDF and CDF. The possible values of (
x,y
) are: (0,0) ,(0,1),(1,0) and (1,1).
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y
1
0
FX,Y
(x,y)
x
CDF
(^
)^
(^
)^
(^
∑∑=
=
−
−
=
=
=
N n
M m
m
n
m
n
Y X^
y y u x x u y Y x X P y x F
1
1
,^
]
,
[
,
) 1
( ) 1
( 1 4 ) ( ) 1
( 1 4 ) 1
( ) ( 1 4 ) ( ) ( 1 4
− − + − + − +
=^
y u x u y u x u y u x u y u x u
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Example 5:
Given the joint PDF
(^
)^
,
x
b
y
x
f^
Y X^
elsewhere
(^2) x
y
x^
Find b for a valid PDF and the marginal PDF’s
f
(X
x ) and f
Y
( y
∞ ∫ ∫∞−
∞ ∞−
=
1
) , ( ,^
dy dx y x
f^
Y X
(^2) /
1 0 2
)
(^2) (
4
1 0
2
1 0
2 2
b
bx
dx x
bx
dy
dx bx
x x
=
=
=
⇒
∫
∫^
∫ −
Integrate over the set of possible (
x,y
).
1
0
x
y
Thus b=2.
y=x
2
For 0<
20
x<
1, we have
(integrate y over blue vertical bar--boundaries
are y=-x
2 and y=x
2 )
3 4
2
) (
2 2
x
dy x
x f
x x
X^
=
=
∫ −
Solution:
{Y=
y } {X=
x }
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