[Math] Random Walk 1D – Variance calculation

Question

I'm trying to solve the following problem:

Suppose that a particle starts at the origin of the real line and moves along the line in jumps of one unit
(where jumps are independent). For each jump, the probability is p that it jumps one unit to the right and 1 − p that it
jumps one unit to the left. Let Xn be the position of the particle after n jumps.

I've calculated the expected value of $X_n$ using $Y_i$ for $1 \leq i \leq n $ and $X_n=\sum_{i=1}^nY_i$
so $E(X_n)=n(2p-1)$ but I'm struggling to calculate $V(X_n)$ !

Any help is appreciated 🙂

Answer 1

Since the jumps are independent and identically distributed, it follows that $$ \mathrm{var}(X_n)=\sum_{i=1}^n\mathrm{var}(Y_i)=n\mathrm{var}(Y_1) $$ so you just need to compute the variance of one of the jumps.