Lecture 4
Mariana OlveraCravioto
UC Berkeley
[email protected]
January 26th, 2017
IND ENG 173, Introduction to Stochastic Processes
Lecture 4
1/12
More on conditional expectation and variance
I
The conditional expectation and the conditional variance of
X
given
Y
=
y
are functions of
y
:
g
(
y
) =
E
[
X

Y
=
y
]
and
h
(
y
) = Var(
X

Y
=
y
)
I
E
[
X

Y
]
,
g
(
Y
)
and
Var(
X

Y
)
,
h
(
Y
)
are random variables!
I
E
[
X

Y
]
is called the
conditional expectation of
X
given
Y
, and it
satisfies:
E
[
E
[
X

Y
]] =
E
[
g
(
Y
)] =
X
y
g
(
y
)
p
Y
(
y
)
=
X
y
E
[
X

Y
=
y
]
p
Y
(
y
) =
E
[
X
]
I
Var(
X

Y
)
is called the
conditional variance of
X
given
Y
. However,
E
[Var(
X

Y
)]
6
= Var(
X
)!!!
IND ENG 173, Introduction to Stochastic Processes
Lecture 4
2/12
Total variance formula
I
It is not hard to check that
Var(
X

Y
) =
E
[
X
2

Y
]

(
E
[
X

Y
])
2
I
Taking expectations on both sides gives
E
[Var(
X

Y
)] =
E
[
E
[
X
2

Y
]]

E
[(
E
[
X

Y
])
2
] =
E
[
X
2
]

E
[(
E
[
X

Y
])
2
]
I
Similarly,
Var(
E
[
X

Y
]) =
E
[(
E
[
X

Y
])
2
]

(
E
[
E
[
X

Y
]])
2
=
E
[(
E
[
X

Y
])
2
]

(
E
[
X
])
2
I
Summing the last two equations gives
E
[Var(
X

Y
)] + Var(
E
[
X

Y
]) =
E
[
X
2
]

(
E
[
X
])
2
= Var(
X
)
IND ENG 173, Introduction to Stochastic Processes
Lecture 4
3/12
Random sums
I
On a given day a supermarket has
N
customers. Each customer spends a
random amount of money
X
i
.
I
Let
S
be the total amount of sales on a given day.
S
=
N
X
i
=1
X
i
I
Assume that the
X
i
’s are iid and independent of
N
.
I
Compute
E
[
S
]
and
Var(
S
)
.
IND ENG 173, Introduction to Stochastic Processes
Lecture 4
4/12
Random sums
I
On a given day a supermarket has
N
customers. Each customer spends a
random amount of money
X
i
.
I
Let
S
be the total amount of sales on a given day.
S
=
N
X
i
=1
X
i
I
Assume that the
X
i
’s are iid and independent of
N
.
I
Compute
E
[
S
]
and