In a meeting, $70\%$ of the members favour and $30\%$ oppose a certain proposal. A member is selected at random and we take $X = 0$ if he opposed and $X = 1$ if he is in favour. Find $E(X)$ and ${\mathop{\rm Var}\nolimits} (X)$ .

.: Random variable $X$ has value $X = 0 \Rightarrow P(X) = \cfrac{{30}}{{100}} = \cfrac{3}{{10}}$

$X = 1 \Rightarrow P(X) = \cfrac{{70}}{{100}} = \cfrac{7}{{10}}$

Thus, we have:

$\therefore \quad E(X) = \sum X P(X) = 0 + \cfrac{7}{{10}} = \cfrac{7}{{10}} = 0.7$

Hence, ${\mathop{\rm Var}\nolimits} (X) = E\left( {{X^2}} \right) - {(E(X))^2}$

$= \cfrac{7}{{10}} - {\left( {\cfrac{7}{{10}}} \right)^2} = \cfrac{7}{{10}} - \cfrac{{49}}{{100}} = \cfrac{{70 - 49}}{{100}} = \cfrac{{21}}{{100}} = 0.21$

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