Find the variance of the following distribution.

XP(X) 0 $\frac{5}{{18}}$ $\frac{4}{9}$ $\frac{1}{2}$ $\frac{4}{9}$ $\frac{5}{{18}}$
${{\rm{X}}^{\rm{2}}}{\rm{P}}\left( {\rm{X}} \right)$

0 $\frac{5}{{18}}$ $\frac{8}{9}$

$\frac{3}{2}$ $\frac{{16}}{9}$
$\frac{{25}}{{18}}$

$\therefore$ Variance $= E\left( {{X^2}} \right) - {[E(X)]^2} = \Sigma {X^2}P(X) - {[\Sigma XP(X)]^2}$

$= \left[ {0 + \frac{5}{{18}} + \frac{8}{9} + \frac{3}{2} + \frac{{16}}{9} + \frac{{25}}{{18}}} \right] - {\left[ {0 + \frac{5}{{18}} + \frac{4}{9} + \frac{1}{2} + \frac{4}{9} + \frac{5}{{18}}} \right]^2}$

$= \left[ {\frac{{5 + 16 + 27 + 32 + 25}}{{18}}} \right] - {\left[ {\frac{{5 + 8 + 9 + 8 + 5}}{{18}}} \right]^2}$

$= \frac{{105}}{{18}} - \frac{{35 \cdot 35}}{{18 \cdot 18}} = \frac{{18 \cdot 105 - 35 \cdot 35}}{{18 \cdot 18}}$

$= \frac{{35}}{{18 \cdot 18}}[54 - 35] = \frac{{19 \cdot 35}}{{324}} = \frac{{665}}{{324}}$