Problem:
Suppose that $X$, $Y$, and $Z$ are independent standard normal random variables.
What is the probability of: \begin{equation} P\{ X+Y+Z+\Delta>0 \, | \, Z+\Delta>0, \, Y+Z+\Delta>0 \} \end{equation}
Where $\Delta>0$ is a real number.
I have been trying to solve this problem by using convolutions to get a distribution for a random variable (and then calculate the probability that it is less than zero).
My method has been to address the problem as:
Let $X\sim N(0,1)$, $Y\sim TN(0,1;0,\infty)$, and $Z\sim TN(\Delta,1;0,\infty)$ be independent random variables; then the sum of them: \begin{equation} W = X + Y + Z \end{equation} is distributed according to the pdf: \begin{equation} f_W (x) = \, ? \end{equation}
What is the probability given by:
\begin{equation} \int^{0}_{-\infty}f_W(x)\, dx = \, ? \end{equation}
Note: $N(0,1)$ denotes a standard normal distribution; $TN(\mu,\sigma;a,b)$ is a truncated normal distribution where $a$ and $b$ are the the lower and upper bounds of the truncation respectively.
I know that the distribution of $W$ is given from the convolution of $X$, $Y$, and $Z$:
\begin{equation} f_W = f_X \ast f_Y \ast f_Z \end{equation}
The convolution operator obeys commutativity and associativity.
Here is what I have shown so far:
\begin{equation} f_X \ast f_Y = \sqrt{2}\, \phi\left( \frac{x}{\sqrt{2}} \right) \, \Phi\left( \frac{x}{\sqrt{2}} \right) \end{equation}
\begin{equation} f_X \ast f_Z = \left[\sqrt{2} \, \Phi\left( \Delta \right)\right]^{-1}\phi\left( \frac{x-\Delta}{\sqrt{2}} \right) \, \Phi\left( \frac{x+\Delta}{\sqrt{2}} \right) \end{equation}
And I have also shown (to somewhat abuse notation): \begin{equation} f_{X\ast Y}(x) \ast f_{Z}(x) =\frac{\sqrt{2}}{\Phi(\Delta)} \phi\left( \frac{x-\Delta}{\sqrt{3}} \right) \int_{-\infty}^{x} \phi\left(\frac{ 3u-2(x-\Delta)}{\sqrt{6}} \right) \, \Phi\left( \frac{u}{\sqrt{2}} \right) \, du \end{equation}
Which, as far as I can tell, does not really simplify any further. I've tried integration by parts on the last integral but it does not seem to help very much. My question is two-fold:
- First, is there a "nice" formula for $f_W(x)$? I can't get much further than what I've written above. My intuition is "no" since this is effectively looking for an analytical solution to the volume under a bivariate normal distribution.
- Second, is it possible to calculate the probability that $W$ is less than zero?
As an aside, I have asked a similar question on a related topic.
The question has been edited to reflect that $X$, $Y$, and $Z$ are independent (thanks to @DilipSarwate for pointing this out).