[Math] The variance of a simple random walk/process

Question

I've been trying to wrap my head around this for the past day. Please help!

Let $\epsilon_i = \pm 1$ with equal probabilities independently for $i=1,…,N$.
Then $Z_i = \epsilon_1 + … + \epsilon_i$ is a random walk. $Z_i$ is a random walk process for $i = 1, …, N$.

Why is the variance $var(\epsilon_i) = 1$ and $var(Z_i) = i$ ?

Answer 1

we know the mean $E[\epsilon_j]$=0

so the variance $E[\epsilon_j^2] - (E[\epsilon_j])^2$ is $E[\epsilon_j^2]$ which is 1

due to independence

$var(Z_i)=\sum_{j=1}^i var(\epsilon_j)=\sum_{j=1}^i 1 = i$