A note on the conditional distribution of X when X - y is given

Z. D. Bai and Lawrence A. Shepp

Statistics & Probability Letters, 1994, vol. 19, issue 3, 217-219

Abstract: It is known that the expected conditional variance of the random variable X when X - y is given is strictly positive for at least one value of y if the distribution of X has an absolutely continuous component or has at least two atoms. From this fact, it might be conjectured that this would remain true for any non-degenerate random variable X. However, this is not the case. In this note, we construct a counterexample and show that for every fixed y, with probability one, the conditional distribution of a random variable X with a singularly continuous distribution when X - y is given may be degenerate.

Keywords: Absolutely; continuous; distributions; atoms; conditional; distributions; conditional; variances; non-degenerate; singularly; continuous; distributions (search for similar items in EconPapers)
Date: 1994
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