(Proof of (1.48)) Here we show that if X is a zero-mean rv with a variance σ 2, then the median α satisfies |α|≤ σ .
(a) First show that |α| ≤ σ for the special case where X is binary with equiprobable values at ±σ .
(b) For all zero-mean rv s X with variance σ 2 other than the special case in (a), show that Pr{X ≥ σ } 0.5.
Hint: Use the one-sided Chebyshev inequality of Exercise 1.32.
(c) Show that Pr{X ≥ α} ≥ 0.5. Other than the special case in (a), show that this implies that α <> .
(d) Other than the special case in (a), show that |α| σ . Hint: Repeat (b) and (c) for the rv -X. You have then shown that |α|≤ σ with equality only for the binary case with values ±σ . For rv s Y with a non-zero mean, this shows that |α - Y|≤ σ .
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.