Mathematics Department, UCLA
T. Richthammer
spring 09, midterm 2
May 18, 2009
Midterm 2: Math 170B Probability, Sec. 1
1. Let
X, Y
be
continuous
RVs.
(a) What is the conceptual diﬀerence between
E
(
X

Y
=
y
) and
E
(
X

Y
)?
(b) Give the deﬁnition of
E
(
X

Y
) and show that
E
(
E
(
X

Y
)) =
E
(
X
).
Answer:
(a)
E
(
X

Y
=
y
) is a number for every choice of
y
, whereas
E
(
X

Y
) is a function of
Y
and
thus a RV (which has certain values with certain probabilities).
(b) We have
E
(
X

Y
) =
h
(
Y
), where
h
(
y
) =
E
(
X

Y
=
y
) =
R
xf
X
(
x

Y
=
y
)
dx
=
R
x
f
X,Y
(
x,y
)
f
Y
(
y
)
dx
. So
E
(
E
(
X

Y
)) =
E
(
h
(
Y
)) =
R
h
(
y
)
f
Y
(
y
)
dy
=
R
(
R
x
f
X,Y
(
x,y
)
f
Y
(
y
)
dx
)
f
Y
(
y
)
dy
=
R
xf
X,Y
(
x, y
)
dxdy
=
E
(
X
).
2. You order an item at time 0. It takes a random time
T
until the item is delivered. At a
random time
X
between 0 and
T
you get an email conﬁrming your order. Suppose that
T
is a RV with
E
(
T
) = 6 and
V
(
T
) = 3. Calculate the
expectation
and
variance
of the
time
X
you get the conﬁrmation email in the two cases (a) and (b).
(a) Assume that
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 Winter '10
 RICHTHAMMER
 Normal Distribution, Variance, Probability theory, Exponential distribution, X1

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