Problems

Let \(x_{\low1}\), \(x_{\low2}\), \ldots, \(x_n\) and \(x_{\vphantom {\dot A} n+1}\) be any fixed real numbers. The numbers \(A\) and \(B\) are defined by \[ A = \frac 1 n \sum_{k=1}^n x_{ \low k} \,, \ \ \ B= \frac 1 n \sum_{k=1}^n (x_{\low k}-A)^2 \,, \ \ \ \] and the numbers \(C\) and \(D\) are defined by \[ C = \frac 1 {n+1} \sum\limits_{k=1}^{n+1} x_{\low k} \,, \ \ \ D = \frac1{n+1} \sum_{k=1}^{n+1} (x_{\low k}-C)^2 \,. \]

1. Express \( C\) in terms of \(A\), \(x_{\low n+1}\) and \(n\).
2. Show that $ \displaystyle B= \frac 1 n \sum_{k=1}^n x_{\low k}^2 - A^2\,\(.
3. Express \)D \( in terms of \)B\(, \)A\(, \)x_{\low n+1}\( and \)n$. Hence show that \((n + 1)D \ge nB\) for all values of \(x_{\low n+1}\), but that \(D < B\) if and only if \[ A-\sqrt{\frac{(n+1)B}{n}} < x_{\low n+1} < A+\sqrt{\frac{(n+1)B}{n}}\,. \]