Mathematics - Statistics Question with Solution | TestHub

Outcomes are $\left(\frac{1}{2}-d\right),\left(\frac{1}{2}-d\right),0\dots ,10$ times, $\frac{1}{2},\frac{1}{2}$, $\dots ,10$ times, $\frac{1}{2}+d,\frac{1}{2}+d,\dots ,10$ times $$\text{ Mean }=\frac{1}{30}\left(\frac{1}{2}\times 30\right)=\frac{1}{2}$$ Variance of the outcomes is, $$\begin{array}{rl}{\sigma }^2& =\frac{1}{30}\Sigma x_i^2-(\bar{x}{)}^2\\ & =\frac{1}{30}\left[{\left(\frac{1}{2}-d\right)}^2\times 10+{\left(\frac{1}{2}\right)}^2\times 10+{\left(\frac{1}{2}+d\right)}^2\times 10\right]-\frac{1}{4}\\ \Rightarrow & \frac{4}{3}=\frac{1}{30}\left[30\times \frac{1}{4}+20d^2\right]-\frac{1}{4}\\ \Rightarrow & \frac{4}{3}=\frac{1}{4}+\frac{2}{3}{d}^2-\frac{1}{4}\\ \Rightarrow & d^2=2\Rightarrow |d|=\sqrt{2}\end{array}$$