91Ó°ÊÓ

X is a random variable for which \(E\left( X \right) = \mu \), \(Var\left( X \right) = {\sigma ^2}\), and c is an arbitrary constant.

\(\begin{aligned}{c}E\left\{ {{{\left( {X - c} \right)}^2}} \right\} &= E\left( {{X^2} - 2Xc + {c^2}} \right)\\ &= E\left( {{X^2}} \right) - 2cE\left( X \right) + {c^2} \ldots \ldots \ldots \left( 1 \right)\end{aligned}\)

Now, we know that \(\begin{aligned}{c}Var\left( X \right) &= E\left( {{X^2}} \right) - {\left\{ {E\left( X \right)} \right\}^2}\\ \Rightarrow E\left( {{X^2}} \right) &= Var\left( X \right) + {\left\{ {E\left( X \right)} \right\}^2}\end{aligned}\).

If we use the value of\(E\left( {{X^2}} \right)\)in equation (1), we get,

\(\begin{aligned}{c}E\left\{ {{{\left( {X - c} \right)}^2}} \right\} &= E\left( {{X^2}} \right) - 2cE\left( X \right) + {c^2}\\ &= Var\left( X \right) + {\left\{ {E\left( X \right)} \right\}^2} - 2cE\left( X \right) + {c^2}\\ &= {\sigma ^2} + {\mu ^2} - 2c\mu + {c^2}\\ &= {\left( {\mu - c} \right)^2} + {\sigma ^2}\end{aligned}\)

Hence, \(E\left\{ {{{\left( {X - c} \right)}^2}} \right\} = {\left( {\mu - c} \right)^2} + {\sigma ^2}\).