60% members of a committee favour a certain proposal and 40% members oppose the proposal. A member is selected and let the random variable \(X = 0\) if he opposes and \(X = 1\) if he is in favour. Then the variance of the random variable X is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{6}{25}\)
X takes the value 1 with probability 0.6 and the value 0 with probability 0.4.
\(E(X) = 1 \times 0.6 + 0 \times 0.4 = 0.6\)
\(E(X^2) = 1^2 \times 0.6 + 0^2 \times 0.4 = 0.6\)
By the variance formula \(\mathrm{Var}(X) = E(X^2) - [E(X)]^2\):
\(\mathrm{Var}(X) = 0.6 - 0.36 = 0.24\)
\(\displaystyle = \frac{24}{100} = \frac{6}{25}\)
Why the other options are incorrect
Option (1) \(\displaystyle\frac{3}{5}\) — Incorrect. This is the mean E(X) = 0.6, not the variance.
Option (2) \(\displaystyle\frac{6}{5}\) — Incorrect. It is greater than 1, but X lies between 0 and 1, so \(\mathrm{Var}(X) = pq\) cannot exceed \(\frac{1}{4}\).
Option (4) \(\displaystyle\frac{3}{10}\) — Incorrect. \(\frac{3}{10} = 0.3\), but the variance is 0.24.