107
If X be a random variable taking values $x_1, x_2, x_3, \ldots, x_{\mathrm{n}}$ with probabilities $\mathrm{P}_1, \mathrm{P}_2, \mathrm{P}_3, \ldots, \mathrm{P}_{\mathrm{n}}$, respectively. Then, $\operatorname{Var}(x)$ is equal to ......... .
Explanation
$$\begin{aligned} \operatorname{Var}(X) & =E(X)^2-[E(X)]^2 \\ & =\sum_{i=1}^n X^2 P(X)-\left[\sum_{i=1}^n X P(X)\right]^2 \\ & =\Sigma P_i x_i^2-\left(\Sigma P_i x_j\right)^2 \end{aligned}$$
108
Let $A$ and $B$ be two events. If $P(A / B)=P(A)$, then $A$ is ........... of $B$.
Explanation
$$\begin{array}{lr} \because & P(A / B)=\frac{P(A \cap B)}{P(B)} \\ \Rightarrow & P(A)=\frac{P(A \cap B)}{P(B)} \\ \Rightarrow & P(A) \cdot P(B)=P(A \cap B) \end{array} $$ So, $A$ is independent of $B$.