A random variable $X$ has the following probability distribution:
If the mean of the random variable $X$ is $1 / 3$, then the variance is

• A

  $\frac{1}{18}$
• Correct answer
  B

  $\frac{5}{18}$
• C

  $\frac{7}{18}$
• D

  $\frac{11}{18}$

Solution

$\because$ Mean $=\sum_{i=1}^n x_i P\left(x_i\right)$ $\Rightarrow \quad \frac{1}{3}=0 \times \frac{25}{36}+1 \times k+2 \times \frac{1}{36}$ $\Rightarrow \quad \frac{1}{3}=k+\frac{1}{18}$ $\therefore \quad k=\frac{1}{3}-\frac{1}{18}=\frac{6-1}{18}=\frac{5}{18}$ Now, variance $=\sum_{i=0}^n x_i^2 P_i(x)-(\text { mean })^2$ $=(0)^2 \times \frac{25}{36}+(1)^2 \times k+(2)^2 \times \frac{1}{36}-\frac{1}{9}$ $=0+k+\frac{1}{9}-\frac{1}{9}=k=\frac{5}{18}$ Hence, variance $=\frac{5}{18}$

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