Question

For randomly moving particles, why is the expectation value of [the x component of the velocity]^2=(1/3)the expectation value of the velocity^2?

Answer 1

\[\left \langle v_x^2\right \rangle=\frac{1}{3}\left \langle v^2\right \rangle\]

Answer 2

I thought of a way to find \[\left | v_x^2 \right |\], but have no clue whether it is correct or how to find the expectation value for that.

Answer 3

\[v_x=\frac{\text{No. of possible directions*x-component for each of these directions}}{No. of possible directions}\]