An online exam is attempted by 50 candidates out of which 20 are boys. The average marks obtained by boys is 12 with a variance 2. The variance of marks obtained by 30 girls is also 2. The average marks of all 50 candidates is 15. If $\mu$ is the average marks of girls and $\sigma$² is the variance of marks of 50 candidates, then $\mu$ + $\sigma$² is equal to ________________.

$\sigma _b^2$ = 2 (variance of boys)<br><br>n₁ = no. of boys<br><br>${\overline x _b}$ = 12<br><br>n₂ = no. of girls<br><br>$\sigma _g^2$ = 2<br><br>${\overline x _g}$ = $${{50 \times 15 - 12 \times {\sigma _b}} \over {30}} = {{750 - 12 \times 20} \over {30}} = 17 = \mu $$<br><br>variance of combined series<br><br>$$\sigma _{}^2 = {{{n_1}\sigma _b^2 + {n_2}\sigma _g^2} \over {{n_1} + {n_2}}} + {{{n_1}.\,{n_2}} \over {{{({n_1} + {n_2})}^2}}}{\left( {{{\overline x }_b} - {{\overline x }_g}} \right)^2}$$<br><br>$$\sigma _{}^2 = {{20 \times 2 + 30 \times 2} \over {20 + 30}} + {{20 \times 30} \over {{{(20 + 30)}^2}}}{(12 - 17)^2}$$<br><br>$\sigma$² = 8<br><br>$\Rightarrow$ $\mu$ + $\sigma$² = 17 + 8 = 25