There are $n$ random vectors of the form ( $y_{1}, y_{2}, y_{3}$ ), where exactly one random coordinate is equal to 1, and the rest are equal to 0.

They are added together. The result is a random vector $\boldsymbol{a}$ with coordinates ( $Y_{1}, Y_{2}, Y_{3}$ ).

a) Find the expected value of the random variable $\boldsymbol{a}^{2}$.

b) Prove that $|\boldsymbol{a}| \geq \frac{n}{\sqrt{3}}$.