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**Jame** We are given that the joint density of X and Y is

$\displaystyle f(x,y) = \begin{cases} & .5 \; \text{if } \; 0 \leq x \leq .5 \; \;\text{and}\; 0 \leq y \leq .5 \\ & 1.25 \; \text{if } \; 0 \leq x \leq .5 \; \;\text{and}\; .5 \leq y \leq 1 \\ & 1.5 \; \text{if} \; .5 \leq x \leq 1 \; \;\text{and}\; 0 \leq y \leq .5 \\ & .75 \; \text{if} \; .5 \leq x \leq 1 \; \;\text{and}\; .5 \leq y \leq 1 \\ \end{cases}$

And are asked to find the conditional distribution of Y given X = .75

the numerator $\displaystyle f(.75,y)$ is

1.5 if $\displaystyle 0 \leq y \leq 5$

.75 if $\displaystyle .5 \leq y \leq 1$

The thing that troubles me is calculating the marginal density of X at .75

$\displaystyle f_x(.75) = \int_{0}^{.5} 1.5 \;\text{d}y \;+\; \int_{.5}^{1} .75 \;\text{d}y \; = 1.125$

We have the event X=.75, which should have probability 0, but when we calculate it's marginal density at .75 we get a non zero number which is (even worse) greater than 1.

Is this something that just happens when we condition on events with probability 0? In general, how do conditional distributions even work in the continous case when we coondition on an event probability 0?