EXERCISES IN STATISTICS
Series A, No. 3
1. If
f
(
x
) =
a
+
bx
2
;
0
≤
x
≤
1, determine
a
and
b
such that
E
(
x
) =
3
4
.
Answer:
The condition that
f
(
x
)
dx
= 1, which is fulfilled by any probability
density function, can be expressed as
1
0
(
a
+
bx
2
)
dx
=
ax
+
bx
3
3
1
0
= 1
.
(1)
The condition that
E
(
x
) = 3
/
4 can be written as
1
0
x
(
a
+
bx
2
)
dx
=
ax
2
2
+
bx
4
4
1
0
=
3
4
.
(2)
From (1) we have
a
+
b
3
= 1
⇐⇒
3
a
+
b
= 3
,
and, from (2), we have
a
2
+
b
4
=
3
4
⇐⇒
2
a
+
b
= 3
.
It follows that
a
= 0,
b
= 3.
2. Let
f
(
x
) = 1;
0
≤
x
≤
1. Find
(a) the mean and variance of
x
,
(b) the mean and variance of
x
2
.
Answer:
E
(
x
) =
1
0
xdx
=
x
2
2
1
0
=
1
2
E
(
x
2
) =
1
0
x
2
dx
=
x
3
3
1
0
=
1
3
V
(
x
) =
E
(
x
2
)
− {
E
(
x
)
}
2
=
1
3
−
1
4
=
1
12
.
Next
E
(
x
4
) =
1
0
x
4
dx
=
x
5
5
1
0
=
1
5
,
1
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SERIES A No.3 : ANSWERS
V
(
x
2
) =
E
(
x
4
)
−
E
(
x
2
)
2
=
1
5
−
1
9
=
4
45
.
3. The probability that
x
buses will pass me before the 106 arrives is given
by
f
(
x
) =
1
4
(
3
4
)
x
. What is the probability that another five buses will
pass me before the 106 arrives, given that three have already passed?
Answer:
The buses which stop here are the 106 the 253 and the 277. We
have
P
(106) =
1
4
and
P
(277
∪
254) = 1
−
P
(106) =
3
4
as the probabilities for
the next arrival. Moreover, these probabilities are independent. Therefore the
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 Spring '12
 D.S.G.Pollock
 Probability, Variance, Probability theory, series A

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