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 $V a r (X)$ .

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Solution

It is given that $P (X = 0) = 30$ % = \displaystyle\frac{30}{100} = 0.3$
$P (X = 1) = 70$ % = \displaystyle\frac{70}{100} = 0.7$
Therefore, the probability distribution is as follows.
Then, $E (X) = \sum X_{i} P (X_{i})$
$= 0 \times 0.3 + 1 \times 0.7$
$= 0.7$
$E (X^{2}) = \sum X_{i}^{2} P (X_{i})$
$= 0^{2} \times 0.3 + {(1)}^{2} \times 0.7$
$= 0.7$
It is known that, $V a r (X) = E (X^{2}) - {[E (X)]}^{2}$
$= 0.7 - {(0.7)}^{2}$
$= 0.7 - 0.49$
$= 0.21$

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