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I’m looking for an online source/article/lecture notes/ text book that would contain a detailed/rigorous discussion/explanation/proof of the following result, which was used in Conditional normal distribution

$P(X+Y

Many thanks

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Using indicator function $\chi$: $ \begin{eqnarray} \mathbb{P}(X+Y

The first line is the definition of probability. Since $\chi_{y in the region $y>b$, we replaced the upper bound of integration w.r.t. $y$ variable with $b$, similarly, $\chi_{x+y is zero for $x>a-y$.

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    just fixed a little typo in your answer – 2011-09-29
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First of all, how this formula can be derived. Suppose, distributions of both $X,Y$ has continuous densities $g_X,g_Y$. Then $ P(X+Y by the Law of total probability. Note that $P(X+Y if $y\geq b$ and $ P(X+Y if $y so $ \int\limits_\mathbb R P(X+Y which is the formula in your post.

Note: although I've cited the Law of total probability from Wikipedia you may be interested in the proof of it. I've found it in these lecture notes. They are in *.ps so you may need some software to read it. Also, I would advise you to read the serious book in probability about this topic. Durrett's book is very nice.

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    That's why we don't multiply by its probability but multiply by the density of $Y$. I would say that for this case the notion of conditional probability is define through the joint distribution rather than through a fraction of two probabilities. You can think about it like a limit case, but for the better understanding I still recommend you to read at least lecture notes I've cited. – 2011-09-29