Let (left{X_{1}, X_{2}, ldots ight}) be a homogeneous discrete-time Markov chain with state space (mathbf{Z}={0,1, ldots, n})

Let \(\left\{X_{1}, X_{2}, \ldots\right\}\) be a homogeneous discrete-time Markov chain with state space \(\mathbf{Z}=\{0,1, \ldots, n\}\) and transition probabilities

\[p_{i j}=P\left(X_{k+1}=j \mid X_{k}=i\right)=\left(\begin{array}{c} n \\ j \end{array}\right)\left(\frac{i}{n}\right)^{j}\left(\frac{n-i}{n}\right)^{n-j} ; i, j \in \mathbf{Z}\]

Show that \(\left\{X_{1}, X_{2}, \ldots\right\}\) is a martingale. (In Genetics, this martingale is known as the Wright-Fisher model without mutation.)