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We are given two independent random variables, first is $X$ which follows a normal distribution $(1,1)$ and second is $Y$ which also follows a normal distribution $(2,1)$. We are given $Z=X+Y$ and asked to find $\mathbb P(Z=0)$.

Firstly is the answer for this $0$? If it is then I don't quite no the proper way to get to this as I guessed it to be zero simply because for a continuous r.v, $\mathbb P(X=x)=0$. So if $Z$ defines any operation on $X$ and $Y$ (not just $X+Y$ in this case) which both follow a continuous distribution, the answer to that would be $0$ as again $\mathbb P(X=x)=0$.

If the answer is not zero, then how exactly would I go about answering this?

Another quick question, if a variable is defined in terms two other variables that are e.g continuous then would that third variable e.g Z in this case also be continuous , and does it analogously apply for the discrete case as well?

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No, the answer is zero as you said, due to continuity. But it seems weird that an exercise would ask that... Is it perharps Z<0 or something like that, that is asked? – Jimmy R. Mar 13 '14 at 22:44
Thanks for your reply, but they actually asked us to find probability that Z=0 – user134785 Mar 13 '14 at 22:46
No, I thought about it again, it is zero due to continuity. – Jimmy R. Mar 13 '14 at 22:48

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