
3.34. Random variable X is uniformly distributed over the range (−1, 1) and Y = X
2
. Find the
covariance.
Solution: The marginal probability density is
f
X
(x) =
1
2
, − 1 ≤ x ≤ 1.
The mean values are
E {X} =
1
−1
xf
X
(x) dx = 0
E {Y } =
1
−1
x
2
f
X
(x) dx =
1
3
.
The second joint moment is
E {XY } = E
$
X · X
2
%
= E
$
X
3
%
=
1
−1
x
3
f
X
(x) dx = 0.
Then,
Cov {X, Y } = E {XY } −E {X}E {Y } = 0.
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