# Finding an expression for the CDF of sum of two random variables $X, Y$ conditioned on the value of one variable $Y$: find $P(X + Y < c | Y = b)$ [closed]

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?

• Note that $$P(X+Y<c|Y=b)=\frac{P(X+Y<c\cap Y=b)}{P(Y=b)}$$ Dec 14, 2020 at 8:01
• Isn't that just the definition of conditional probability? How do I proceed?
– John
Dec 14, 2020 at 13:09
• Ah, you mean the numerator isn't just $F(c)$.. how do i write it then as a function of the cdf?
– John
Dec 14, 2020 at 13:12
• @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 $P(Y = b) = 0$.
– John
Dec 14, 2020 at 14:11
• 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.
– whuber
Dec 14, 2020 at 16:47