Iām 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
Many thanks
Iām 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
Many thanks
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 $x>a-y$.
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.