Another formula to find the variance of a random variable We know that,
Var (X) =
2
1
(
μ)
(
)
n
i
i
i
x
p x
=
−
∑
=
2
2
1
(
μ
2μ
) (
)
n
i
i
i
i
x
x
p x
 
 
 
 
=
2
2
1
1
1
(
)
μ
(
)
2μ
(
)
n
n
n
i
i
i
i
i
i
i
i
x
p x
p x
x p x
=
=
=
+
−
∑
∑
∑
=
2
2
1
1
1
(
) μ
(
)
2μ
(
)
n
n
n
i
i
i
i
i
i
i
i
x
p x
p x
x p x
=
=
=
+
−
∑
∑
∑
=
2
2
2
1
=1
1
(
) μ
2μ
since
 ( )=1andμ=
( )
n
n
n
i
i
i
i
i
i
i
i
x
p x
p x
x p x
=
=
⎡
⎤
+
−
⎢
⎥
⎣
⎦
∑
∑
∑
=
2
2
1
(
) μ
n
i
i
i
x
p x
=
−
∑
or
Var (X) =
2
2
1
1
(
)
(
)
n
n
i
i
i
i
i
i
x
p x
x p x
=
=
⎛
⎞
−⎜
⎟
⎝
⎠
∑
∑
or
Var (X) = E(X2) – [E(X)]2, where E(X2) =
2
1
(
)
n
i
i
i
x
p x
=∑
Example 28 Find the variance of the number obtained on a throw of an unbiased die Solution The sample space of the experiment is S = {1, 2, 3, 4, 5, 6} Let X denote the number obtained on the throw