If a random variable $X$ denotes the number that appears on the upper face of a die when it is rolled, then $\frac{\text { Variance of } X}{\text { Mean of } X}$ is equal to

• A

  $\frac{7}{2}$
• B

  $\frac{35}{12}$
• Correct answer
  C

  $\frac{5}{6}$
• D

  $\frac{9}{2}$

Solution

$\begin{aligned} \text { Mean }=\mu & =E(x)=\sum_{P_i x_i}=\frac{1}{6}+\frac{2}{6}+\frac{3}{6}+\frac{4}{6}+\frac{5}{6}+\frac{6}{6}=\frac{21}{6} \\ \text { Variance } & =\operatorname{var}(x)=\sigma^2=\sum_{P_i x_i^2}-\mu^2 \\ & =\left(\frac{1}{6}+\frac{4}{6}+\frac{9}{6}+\frac{16}{6}+\frac{25}{6}+\frac{36}{6}\right)-\left(\frac{21}{6}\right)^2 \\ & =\frac{91}{6}-\frac{441}{36}=\frac{546-441}{36}=\frac{105}{36} \\ \therefore \quad \frac{\sigma^2}{\mu} & =\frac{105 / 36}{21 / 6}=\frac{105}{36} \times \frac{6}{21}=\frac{5}{6}\end{aligned}$

Hence, the correct answer is option: 'C'.