1
Welcome Back Our Friend: Expectation
•
Recall expectation for discrete random variable:
•
Analogously for a continuous random variable:
•
Note: If X always between
a
and
b
then so is E[X]
More formally:
x
x
p
x
X
E
)
(
]
[
dx
x
f
x
X
E
)
(
]
[
b
X
E
a
b
X
a
P
]
[
1
)
(
then
if
Generalizing Expectation
•
Let
g
(
X
,
Y
) be real-valued function of two variables
•
Let X and Y be discrete jointly distributed RVs:
•
Analogously for continuous random variables:
y
x
Y
X
y
x
p
y
x
g
Y
X
g
E
)
,
(
)
,
(
)]
,
(
[
,
dy
dx
y
x
f
y
x
g
Y
X
g
E
y
x
Y
X
)
,
(
)
,
(
)]
,
(
[
,
Expected Values of Sums
•
Let g(X, Y) = X + Y.
Compute E[g(X, Y)] = E[X + Y]
•
Generalized:
Holds regardless of dependency between
X
i
’s
dy
dx
y
x
f
y
x
Y
X
E
y
x
Y
X
)
,
(
)
(
]
[
,
dy
dx
y
x
f
y
dx
dy
y
x
f
x
y
x
Y
X
x
y
Y
X
)
,
(
)
,
(
,
,
dy
y
f
y
dx
x
f
x
y
Y
x
X
)
(
)
(
]
[
]
[
Y
E
X
E
n
i
i
n
i
i
X
E
X
E
1
1
]
[
Tie Me Up! : Bounding Expectation
•
If random variable X ≥
a
then
E[X] ≥
a
Often useful in cases where
a
= 0
But, E[X] ≥
a
does not
imply X ≥
a
for all X = x
o
E.g., X is equally likely to take on values
-1 or 3.
E[X] = 1.
•
If random variables X ≥ Y
then
E[X] ≥ E[Y]
X ≥ Y
X
– Y ≥ 0
E[X
– Y] ≥ 0
Note: E[X
–
Y] = E[X] + E[-Y] = E[X]
–
E[Y]
Substituting: E[X]
– E[Y] ≥ 0
E[X] ≥ E[Y]
But, E[X] ≥ E[Y] does
not
imply X ≥ Y for all X = x, Y = y
]
[
1
)
(
X
E
a
X
a
P
then
if

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