The Conditional Distribution of X Given X = Y can be Almost Anything!

Suppose that X and Y are independent absolutely continuous random variables. We consider the problem of deciding what is an appropriate choice for the conditional density of X given X = Y. It has been earlier noted [see, for example, Rao (1993)] that ambiguous responses to this question are possible. In the present note we argue that the situation is actually worse than this might suggest. In fact the conditional density of X given X = Y is essentially arbitrary. It can be whatever you wish!