Pressure of a Gas and average velocities in each direction I am learning to derive the equation of pressure of the gas which is given at the bottom of following photo.

Just want the explanation of the statement "Because there are many molecules and because they are all moving in random direction, the average values of the squares of their velocity components are equal, so that $(v_x)^2=\frac{v^2}{3}$"
I think author wants to say,
$$\frac{(v_{1x})^2 + (v_{2x})^2 + ... + (v_{Nx})^2}{N}=\frac{(v_{1y})^2 + (v_{2y})^2 + ... + (v_{Ny})^2}{N}=\frac{(v_{1z})^2 + (v_{2z})^2 + ... + (v_{Nz})^2}{N}$$
Which implies:
$$(v_{1x})^2 + (v_{2x})^2 + ... + (v_{Nx})^2 = (v_{1y})^2 + (v_{2y})^2 + ... + (v_{Ny})^2 = (v_{1z})^2 + (v_{2z})^2 + ... + (v_{Nz})^2 ---(A)$$
The problem is I can't understand how the statement is logical. How is it possible to assume and say something which is shown by equation A. I don't understand why the sums of the squares of the velocity components of all the molecules would be equal to each other. Or whether I have misunderstood the author. 
 A: The author is saying that, $v^2 = v_x^2 + v_y^2 +v_z^2$, and because the velocities are random, then $\langle v_x^2 \rangle = \langle v_y^2 \rangle = \langle v_z^2 \rangle$, and thus, $$v^2 = 3 \langle v_x^2 \rangle,$$ where the angled brackets denote averages.
The averages can be interpreted both as ensemble (over the different particles) and over time, i.e. both that $\langle v^2_{i,x}(t_1) \rangle = \langle v^2_{i,x}(t_2) \rangle$, and that $\langle v^2_{i,x}\rangle = \langle v^2_{j,x} \rangle = \langle v^2_{j,y} \rangle$.  This follows from the premistes of equipartition (in dimensions and between particles) and an equilibrium configuration.
A: You're making a simple matter complex. It's simply stated that average of each velocity component of the gas molecules are same with no preference to one component over the other. If you turn the gas container upside down all velocity components are on average same in equilibrium. I.E. $$\langle v_x\rangle =\langle v_y\rangle =\langle v_z\rangle  $$ so $\langle v^2_x\rangle $ can be taken as 1/3rd of the total velocity squared.
