Calculate the bias of the estimator

Question

given is $c(t,s) = E[(X(t) - \mu(t))(X(s) - \mu(s))]$ and the estimator $\hat{c}(t,s) = \frac{1}{n} \sum\limits_{i = 1}^n (X_i(t) - \hat{\mu}(t))(X_i(s) - \hat{\mu}(s))$ with $\mu(t) = E[X(t)]$ with $X_1,X_2, ..., X_n$ iid. I have to show that $E[\hat{c}(t,s)] = \frac{n}{n-1}c(t,s)$. How do I get this the factor $\frac{n}{n-1}$? Replacing $\hat{\mu}$ by $\mu$ has a negligible effect, so $E[\hat{c}(t,s)] = E[\frac{1}{n} \sum\limits_{i = 1}^n (X_i(t) - \hat{\mu}(t))(X_i(s) - \hat{\mu}(s))] = E[\frac{1}{n} \sum\limits_{i = 1}^n (X_i(t) - \mu(t))(X_i(s) - \mu(s)) = \frac{1}{n} \sum\limits_{i = 1}^n E[(X_i(t) - \mu(t))(X_i(s) - \mu(s))] = \frac{1}{n} n E[(X_1(t) - \mu(t))(X_1(s) - \mu(s))] = c(t,s).$