We stressed the importance of the mean of a rv X in terms of its associ- ation with the sample average via the WLLN. Here we show that in essence the WLLN allows us to evaluate the entire CDF, say FX (x) of X via sufficiently many independent sample values of X.
(a) For any given y, let Ij(y) be the indicator function of the event {Xj ≤ y}, where X1, X2, ... , Xj, ... are IID rv s with the CDF FX (x). State the WLLN for the IID rv s {I1(y), I2(y), .. .}.
(b) Does the answer to (a) require X to have a mean or variance?
(c) Suggest a procedure for evaluating the median of X from the sample values of X1, X2, ... . Assume that X is a continuous rv and that its PDF is positive in an open interval around the median. You need not be precise, but try to think the issue through carefully.
What you have seen here, without stating it precisely or proving it is that the median has a law of large numbers associated with it, saying that the sample median of n IID samples of a rv is close to the true median with high probability.
Text Book: Stochastic Processes: Theory for Applications By Robert G. Gallager.