Assume a random walker that starts at position $(x_0,y_0)=(0,0)$ and performs $N$ steps. Each step is either up, down, left or right and is of length 1. The walker decides uniformly at random among these 4 options where to go next.
Let $(x_N,y_N)$ be his final position after $N$ steps and the quantity $d=\sqrt{x_N^2+y_N^2}$ is his distance from the initial position (origin). I am trying to show the following expression for the expectation of its square $$E[d^2]=N$$ What I did is \begin{eqnarray*} E[d^2]&=&E[\sqrt{x_N^2+y_N^2}^2]\\ &=&E[x_N^2+y_N^2]\\ &=&E[x_N^2]+E[y_N^2]\\ &=&2E[x_N^2]\qquad\text{since the variables are i.i.d.}\\ &=&2\sum\limits_{i=0}^N E[x_i^2]\qquad\text{since }X_i\text{ are i.i.d. and zero-mean}\\ &=&2\sum\limits_{i=1}^N E[x_i^2]\\ &=&2\sum\limits_{i=1}^N (Var(X_i)+E^2[x_i])\\ &=&2\sum\limits_{i=1}^N Var(X_i)\\ &=&2NVar(X_1)\\ &=&2N\frac{1}{2}(1-\frac{1}{2})\\ &=&\frac{N}{2} \end{eqnarray*}
but this doesn't match what I am trying to prove and I cannot find where I am wrong.
We have $x_N=\sum_{i=1}^NX_i$ and $y_N=\sum_{i=1}^NY_i$ where the $(X_i,Y_i)$ take values in $\{\langle1,0\rangle,\langle-1,0\rangle,\langle0,1\rangle,\langle0,-1\rangle\}$ with uniform distribution.
$\mathsf Ed=\mathsf Ex_N^2+\mathsf Ey_N^2=2\mathsf Ex_N^2=2\mathsf \sum_{i=1}^N\sum_{j=1}^N\mathsf EX_iX_j=2N(N-1)\mathsf EX_1X_2+2N\mathsf EX_1^2$
Here $\mathsf EX_1X_2=\mathsf EX_1\mathsf EX_2=0\cdot0=0$ and $\mathsf EX_1^2=\frac14\cdot1^2+\frac14\cdot(-1)^2=\frac12$
So we end up with $$\mathsf Ed^2=N$$