If we take 8 identical slips of paper and write the number 0 on one of them, the number 1 on three of the slips, the number 2 on three of the slips and the number 3 on one of the slips. These slips are folded, put in a box and roughly mixed. One slip is drawn at random from the box. If X is the random variable denoting the number written on the drawn slip, the variance of X is:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 3/4
Out of 8 slips, one shows 0, three show 1, three show 2 and one shows 3:
\(P(X=0) = \frac{1}{8},\quad P(X=1) = \frac{3}{8},\quad P(X=2) = \frac{3}{8},\quad P(X=3) = \frac{1}{8}\)
Mean:
\( \displaystyle E(X) = \frac{0 + 3 + 6 + 3}{8} = \frac{12}{8} = \frac{3}{2} \)
Mean of the squares:
\( \displaystyle E(X^2) = \frac{0 + 3(1) + 3(4) + 1(9)}{8} = \frac{24}{8} = 3 \)
By the variance formula \(\mathrm{Var}(X) = E(X^2) - [E(X)]^2\):
\( \displaystyle \mathrm{Var}(X) = 3 - \frac{9}{4} = \frac{3}{4} \)