Consider a point on a plane starting at the origin ($X_0, Y_0$ = $0,0$). At each discrete step, the point moves a distance of one unit in four possible directions, with an equal probability of 1/4 for each possible move. Find E[$X_N^2 + Y_N^2$] for all nonnegative integers N. The hint that is given is that this is a variance problem.
This is where I am so far: P(left = -1) = 1/4 P(right = +1) = 1/4 P(no step) = 1/2
P(up = +1) = 1/4 P(down = -1) = 1/4 P(no step) = 1/2
For each axis separately, say the X-axis, I'm at the following: E(x) = 0, Var(x) = N/2 after N steps
But how do I go about combining this for two axises?
We can write $X_i = X_{i-1} + Z_i$, $i\in \mathbb N$ where the $Z_i$'s are idd with $P(Z_i = \pm 1) = 1/4$ and $P(Z_i = 0) = 1/2$. Then by iterating back to $X_0$, $$ X_N = X_{N-1} + Z_N = X_{N-2} + Z_{N-1} + Z_N = \ldots = \underbrace0_{X_0} + \sum_{i = 1}^N Z_i. $$ Remember that $E[Z^k] = \sum_n n^k P(Z = n)$ so you should get that \begin{align} E[X_N^2] &= E[(Z_1 + \ldots + Z_N)(Z_1 + \ldots + Z_N)] \\&= E[Z_1^2+ \ldots + Z_N^2] + \sum_{i = 1;i\ne j}E[Z_iZ_j] \\&= E[NZ^2] + 0 \\&= N\cdot 1/2. \end{align}
Now, $X_i \stackrel d= Y_i$ by symmetry so also $E[Y_N] = 0$ and $E[Y_N^2] = N/2$.
Then by linearity of the expected value,
$$ E[X_N^2+Y_N^2] = E[X_N^2] + E[Y_N^2] = N/2+N/2 = N. $$