Mathematics - Statistics Question with Solution | TestHub

Since, Mean $=\frac{\sum {\mathrm{f}}_{\mathrm{i}}{\mathrm{x}}_{\mathrm{i}}}{\sum {\mathrm{f}}_{\mathrm{i}}}$ where ${\mathrm{x}}_{\mathrm{i}}$ are observations with frequencies ${\mathrm{f}}_{\mathrm{i}},\mathrm{i}=1,2,$ $\dots \dots \dots$ The required mean is given by $$\begin{array}{rl}\overline{\mathrm{X}}& =\frac{0.1+1.{}^{\mathrm{n}}{\mathrm{C}}_1+2{}^{\mathrm{n}}{\mathrm{C}}_2+\dots \dots +\mathrm{n}.{}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}}{1+{}^{\mathrm{n}}{\mathrm{C}}_1+{}^{\mathrm{n}}{\mathrm{C}}_2+\dots +{}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{n}}}\\ =& \frac{{\sum }_{\mathrm{r}=0}^{\mathrm{n}}{\mathrm{r}}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}}{{\sum }_{\mathrm{r}=0}^{\mathrm{n}}{}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}}=\frac{{\sum }_{\mathrm{r}=1}^{\mathrm{n}}\mathrm{r}\cdot \frac{\mathrm{n}}{\mathrm{r}}{}^{\mathrm{n}-1}{\mathrm{C}}_{\mathrm{r}-1}}{{\sum }_{\mathrm{r}=0}^{\mathrm{n}}{}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}}\\ =& \frac{\mathrm{n}\sum {\sum }^{\mathrm{n}-1}{\mathrm{C}}_{\mathrm{r}-1}}{{\sum }_{\mathrm{r}=0}^{\mathrm{n}}{}^{\mathrm{n}}{\mathrm{C}}_{\mathrm{r}}}=\frac{\mathrm{n}.2^{\mathrm{n}-1}}{2^{\mathrm{n}}}=\frac{\mathrm{n}}{2}\end{array}$$