If \(x\) and \(y\) are statistically independent, then \(E[x y]=E[x] E[y]\). That is, the expected value of the product \(x y\) is equal to the product of the separate mean values. If \(z=x+y\), where \(x\) and \(y\) are statistically independent, show that \(E\left[z^{2}\right]=E\left[x^{2}\right]+E\left[y^{2}\right]+2 E[x] E[y]\).