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
$P(X+Y
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
$P(X+Y
Many thanks
Using indicator function $\chi$:
$$ \begin{eqnarray}
\mathbb{P}(X+Y
The first line is the definition of probability. Since $\chi_{yb$, we replaced the upper bound of integration w.r.t. $y$ variable with $b$, similarly, $\chi_{x+ya-y$.
First of all, how this formula can be derived. Suppose, distributions of both $X,Y$ has continuous densities $g_X,g_Y$. Then
$$
P(X+YLaw of total probability. Note that $P(X+Y
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.