V(n, t, x') = sin [k x' + A (n, t)]
k is a constant
n stands for the nth realisation
A is uniformly distributed in [0, 2pi] and
x' denotes the position vector
How can I show that the mean of V i.e. <V> and the mean of the squares <V^2> are independent of x' and indeed for any <V^n>?
I used <V> = [1/(b-a)] int [a,b] V(n, t, x') dx
where <V> denotes the mean
The Attempt at a Solution
I do not completely understand the question. The average over a complete period or integer multiples of the period is zero for a sine function. The arguments do not affect the average but clearly the interval where you take the average does.
The question implies showing <V>, <V^2> .. <V^n> to be independent for ALL x but being given that A is uniformally distributed in [0, 2pi], does this imply that this is the meant domain of integration for the average?
And how can I generalise for all <V^n>?