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This is related to my other question on renewal processes https://math.stackexchange.com/questions/3947852/renewal-theory-probability-of-residual-lifetime-gamma-t-x-conditioned-on-c

$X, Y$ are continuous random variables, possibly not independent, and we know the $CDF$ of $X + Y$ which is given by $F(c)$. What is $P(X + Y < c | Y = b)$ in terms of $F(c)$?

Attempt:

$P(X + Y < c | Y = b) = \frac{P(X + Y < c)}{P(Y = b)}$. The numerator is just $F(c)$, but how do I find the denominator in terms of the given $CDF$? Another confusion I have is why is the denominator not just $0$ since $Y$ is a continuous random variable?

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    $\begingroup$ Note that $$P(X+Y<c|Y=b)=\frac{P(X+Y<c\cap Y=b)}{P(Y=b)}$$ $\endgroup$
    – gunes
    Dec 14, 2020 at 8:01
  • $\begingroup$ Isn't that just the definition of conditional probability? How do I proceed? $\endgroup$
    – John
    Dec 14, 2020 at 13:09
  • $\begingroup$ Ah, you mean the numerator isn't just $F(c)$.. how do i write it then as a function of the cdf? $\endgroup$
    – John
    Dec 14, 2020 at 13:12
  • $\begingroup$ @gunes Also, aren't we conditioning on an event with $0$ probability in this case? Is that allowed? $Y$ is a continuous random variable, so $P(Y = b) = 0$. $\endgroup$
    – John
    Dec 14, 2020 at 14:11
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    $\begingroup$ Writing $Z=X+Y$ and noting you make no assumptions whatsoever about $(X,Y)$ apart from having continuous marginals, you are asking how to compute conditional probabilities for any pair of continuous random variables $(Y,Z).$ In particular, conditional probabilities do not depend solely on the marginal distribution of $Z$ (if they did, all continuous variables would automatically be independent), which makes your question unanswerable. $\endgroup$
    – whuber
    Dec 14, 2020 at 16:47

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