Support of Continuous Conditional Density Functions (Probability)

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f(x,y) = x + y is the joint probability density function for continuous random variables X and Y. The support of this function is {0 < x < 1, 0 < y < 1}, which means it takes positive values over this region and zero elsewhere.

g(x) = x + (1/2) is the probability density function of X, derived by integrating f(x,y) with respect to y, over 0 < y < 1. Since we "took y out of the equation", the support of this function should be 0 < x < 1.

Question :

What is the support of the conditional density function h(y|x) = f(x,y)/g(x)?

My guess is that this function is well defined and takes non zero values only over region {0 < x < 1, 0 < y < 1} and so this is the support.

But the book I am using states 0 < y < 1 instead. Is it a typo?

 

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I just realized my mistake :

x is fixed since we are conditioning Y on a particular value of X = x. So the 0 < x < 1 is irrelevant. Or rather, the conditional probability density for a given X = x is a function of y only. So only the value of y matters in defining the support.

I feel like an idiot lol.