Midterm 1 Solutions
1. Let
X
and
Y
be random variables with the following joint distribution.
x
y
1
0
1
1
0.1
0.2
0.2
0
0.3
0
0.1
1
0
0
0.1
(a)
Find
P
(
X
≥
0
and
Y
≥
0)
.
Solution.
0 + 0.1 + 0.1 + 0 = 0.2.
(b)
Compute
P
(
X
=
x
)
for all possible values of
x
(i.e.,

1
,
0
, and
1
).
Solution.
x
1
0
1
P
X
(
x
)
0.4
0.2
0.4
Each of the above values is the sum of the entries in the corresponding column in
the original table. (E.g., 0.2 = 0.2 + 0 + 0.)
(c)
What is the variance of
X
?
Solution.
First we compute the mean:
E
(
X
) =
∑
x
xP
X
(
x
) =

1
·
0
.
4+0
·
0
.
2+1
·
0
.
4 = 0. Then Var(
X
) =
∑
x
x
2
P
X
(
x
)

E
(
X
)
2
= (

1)
2
·
0
.
4+0
2
·
0
.
2+1
2
·
0
.
4

0
2
=
0
.
8
.
(d)
Are
X
and
Y
independent?
Solution.
No, because, for example,
P
XY
(0
,
0) = 0, while
P
X
(0) = 0
.
2 and
P
Y
(0) = 0
.
3 + 0 + 0
.
1 = 0
.
4.
2. Let
X
be a random variable with probability density
f
X
(
x
) =
1
2
(1 + 3
x
2
)
,
where 0
< x <
1
.
(a)
What is the mean of
X
?
Solution.
R
1
0
xf
X
(
x
)
dx
=
R
1
0
x
·
1
2
(1 + 3
x
2
)
dx
=
1
2
(
x
2
2
+
3
x
4
4
)
ﬂ
ﬂ
1
0
=
5
8
= 0
.
625
.
(b)
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 Spring '07
 Safarzadeh
 Probability theory, probability density function, Cumulative distribution function, favorable review

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