I am self teaching and i am very stuck on this question
If we have the stochastic process ${X_n : n = 0, 1, 2, . . .} $ which is a Markov chain on the integers with $X_0 = 0$. If $|X_n|$ is even, the conditional probabilities $P(X_{n+1} = i + 1|X_n = i) = 3/4$ and $P(X_{n+1} = i − 1|X_n = i) = 1/4$ are given, whereas if $|X_n|$ is odd, the conditional probabilities are $P(X_{n+1} = i + 1|X_n = i) = 1/4$ and $P(X_{n+1} = i − 1|X_n = i) = 3/4$. Calculate $E[X_n]$ and the variance $Var[X_n] = E[(X_n − E[X_n])^2].$
I know that for asymmetric random walks $X_n = X_0 + \sum ^n_{j=1} Z_j$ and the random variables (the increments) and chosen identically and independently with $P(Z_j = +a)=p $ and, $P(Z_j=-a)=1-p=q$
For the even, i let $p= 3/4 $and $ 1-p =1/4 = q$ and i have that the mean for asymmetric random walks is $E[X_n] =na(p-q)$ and variance $σ_n^2 =na^2[(p+q)-(p-q)^2]$ where $n$ is the number of steps and a is the size of the individual step. In my book its says to assume a= 1 usually.
Would really appreciate the help