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Let $X$ be a continuous random variable with density function $f(\cdot)$. Define $Y = 2X$, another continuous random variable. I would like to determine the conditional density of $f_{Y|X}(y|x)$. It is clear that $f_{Y|X}(y|x) = 0$ for all $y \neq 2x$ and for $y = 2x$, we should probably have an infinite spike. That is, $f_{Y|X}(y|x)$ seems like a Dirac delta function, which is not really a function. Can someone suggest a proper way of handling this situation?

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  • $\begingroup$ It's not a function onto the reals, but it's useful in some kinds of analysis. Anyway, your assessment of the situation seems accurate. Without knowing more about your application, I'm not sure there's a superior approach to the problem. $\endgroup$ – Brian Tung Apr 14 '15 at 17:47
  • $\begingroup$ @BrianTung I was trying to answer math.stackexchange.com/questions/1233203/… and ended up running into this issue. $\endgroup$ – vdesai Apr 14 '15 at 18:33
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As you say, the proper density function for a constant is the dirac delta function. The CDF becomes the indicator function.

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$ f_{Y|X}(y|x)=\delta_{2x}(y) $.

That is, given that $ X=x $, $ Y=2x $ with probability 1.

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