Suppose X and Y are independent standard normal random variables. We are finding the conditional probability distribution function (pdf) of Y given Y=X.
However, there can be different interpretations of the problem. The given fact Y=X can be interpreted as (i) Z=0, where Z=Y-X or (ii) Z=1 where Z=X/Y, or (iii) Z=1 where Z=I(Y=X) (I is the index function)
My professor says that the pdf of Y given Y=X will vary with the interpretation of the condition Y=X. I could not really understand why.
What I could understand till now is this:
I will have different distributions of Z according to the interpretation. In case (i), it will be normal with parameters (0,2). In case (ii), it will be a Cauchy distribution and I don't know about (iii).
Coming back to the point, as I change the interpretation, the $f_Z(z)$ (pdf of random variable Z) changes and so, since the conditional pdf of Y|Z=z is $\frac{f_Y(.)}{f_Z(.)}$, it will also change. Now my question is this:
(1) Will the probability of occurrence of a certain value change with our interpretation?
(2) Will the conditional expectation change with our interpretation?
I think it should not change. But I am unable to reason it out why. Can anyone please help me with this? Thanks in advance!