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
Slips distribution:
0 → 1 slip
1 → 3 slips
2 → 3 slips
3 → 1 slip
Total = 8 slips ⇒ probabilities:
$P(0)=\frac{1}{8},\; P(1)=\frac{3}{8},\; P(2)=\frac{3}{8},\; P(3)=\frac{1}{8}$
Compute mean:
$E(X)=0\cdot\frac{1}{8}+1\cdot\frac{3}{8}+2\cdot\frac{3}{8}+3\cdot\frac{1}{8}$
$=\frac{0+3+6+3}{8}$
$=\frac{12}{8}=1.5$
Compute second moment:
$E(X^{2})=0^{2}\cdot\frac{1}{8}+1^{2}\cdot\frac{3}{8}+2^{2}\cdot\frac{3}{8}+3^{2}\cdot\frac{1}{8}$
$=\frac{0+3+12+9}{8}$
$=\frac{24}{8}=3$
Variance:
$\text{Var}(X)=E(X^{2})-[E(X)]^{2}$
$=3-(1.5)^{2}$
$=3-2.25$
$=0.75$
Variance of X = 0.75