Problem #PRU-66053

Problem

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 others are equal to 0. They are summed up. A random vector a with coordinates $(Y_1,Y_2,Y_3)$ is obtained.

a) Find the mathematical expectation of a random variable $a^2$.

b) Prove that $|a|\ge \frac{1}{3}$.