Link to theorem: http://en.wikipedia.org/wiki/Law_of_the_unconscious_statistician(adsbygoogle = window.adsbygoogle || []).push({});

Suppose Y is a discrete random variable related to X, a continuous random variable by some function r (so Y = r(X) ).

Let A be the following set: A_y = {x ∈ R ; r(x) = y}.

Since Y is discrete, f_Y (y) = P(Y = y) = P(r(X) = y). r(X) = y is equivalent to X ∈ A_y, so f_Y (y) = P(X ∈ A_y) = Sum of all P(X = x) such that x ∈ A_y.

It seems to me that the previous sum is valid for both discrete and continuous X. However, if X is continuous then P(X = x) = 0 for all x ∈ R. Thus X must be discrete, however I can construct a transformation from a continuous variable to a discrete one, so X is not necessarily discrete.

Am I wrong? Can anyone show me my mistake, if there is one? I really would like some clarification on this. Thank you!

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# Question about the rule of the Lazy Statistician - If Y is discrete, w

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