Wald's identities for a random sum

Let \(\xi_1, \ldots, \xi_n, \tau\) be independent random variables, \(\xi_1, \ldots, \xi_n\) have the same distribution, \(\tau\) takes the values \(1, \ldots, n\). Consider the sum of a random number of the random variables \(S_\tau = \xi_1 + \ldots + \xi_\tau\). Show that

1. \(\mathbf{E} S_\tau = \mathbf{E}\tau \cdot \mathbf{E}\xi_1\),

2. \(\mathbf{E}(S_\tau | \tau) = \tau \mathbf{E}\xi_1\),

3. \(\mathbf{D} S_\tau = \mathbf{E}\tau \cdot \mathbf{D}\xi_1 + \mathbf{D}\tau \cdot (\mathbf{E}\xi_1)^2\),

4. \(\mathbf{D}(S_\tau | \tau) = \tau \mathbf{D}\xi_1\).