Let μ = E (X) be the mean of X The variance of X, denoted by Var (X) or 
2
 x
 is
defined as
2
σx
 =
2
1
Var(X)=
(
μ)
(
)
n
i
i
i
x
p x
=
−
∑
or equivalently
2
 x
 = E(X – μ)2
O
1 8
2 8
3 8
P(Y)
O
1 8
2 8
3 8
P(X)
1
2
3
4
1
2
3
4
–1
5
6
(i)
(ii)
© NCERT
not to be republished
PROBABILITY      567
The non-negative number
σx =
2
1
Var(X) =
(
μ)
(
)
n
i
i
i
x
p x
=
−
∑
is called the standard deviation of the  random variable X 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