In the hat check problem (see Example), it was assumed that [math]N[/math] people check their hats and the hats are handed back at random. Let [math]X_j = 1[/math] if the [math]j[/math]th person gets his or her hat and 0 otherwise. Find [math]E(X_j)[/math] and [math]E(X_j \cdot X_k)[/math] for [math]j[/math] not equal to [math]k[/math]. Are [math]X_j[/math] and [math]X_k[/math] independent?