let $ X, N_1, N_2 $ be random Normal Gaussian independent variables with the following distributions: \begin{align*} X \sim \mathcal{N}(\mu_0,\sigma_X ^2 ) \end{align*} \begin{align*} N_1 ,N_2 \sim \mathcal{N}(0,1) \end{align*} we denote: \begin{align*} Y_1 = aX+b+N_1 \end{align*} \begin{align*} Y_2 = aX+b+N_2 \end{align*}
where $a,b \neq 0$ are determinintic known constants.
I'm interested in finding the following:
- The common distribution of $X,Y_1: F_{X,Y_1} $
- The distribution of $X$ given $Y_1: F_{X|Y_1} $
- The common distribution of $X,Y_1, Y_2: F_{X,Y_1,Y_2} $ (where $a=1, b=0$)
- The distribution of $X$ given $Y_1 $ and $Y_2$ : $F_{X|(Y_1,Y_2)} $ (where $a=1, b=0$)
I will start with what I did in the first section:
\begin{align*} F_{X,Y_1}(x,y_1)=\textbf{P}(X \leq x, Y_1 \leq y_1) =\textbf{P}(X \leq x, Y_1 \leq y_1 |N_1=n)\textbf{P}\left(N_1 = n\right) \end{align*} \begin{align*} = \textbf{P}\left(X \leq x, X \leq \frac{y_1 -b-n}{a} |N_1=n \right) \textbf{P}\left(N_1 = n\right) =\textbf{P}\left(X \leq \text{Min} \left(x, \frac{y_1 -b-n}{a}\right) |N_1=n \right) \textbf{P}\left(N_1 = n\right) \end{align*}
\begin{align*} = \textbf{P}\left(N_1 = n\right) \int_{-\infty}^{\text{Min} \left(x, \frac{y_1 -b-n}{a}\right) } \frac{1}{\sqrt{2\pi}\sigma_X }e^{-\frac{(t-\mu_0)^2}{2\sigma_X ^2}}dt \end{align*}
How do I continue from here?
NOTE: I wrote the follow-up questions in order to make this question look at many aspects of calculating normal distributions. That would be better than dividing this into several questions. thus, making the discussion only in one place. once I solve the first section I will get the intuition on solving the follow-up questions and I will add what I've got. Also, I'd be glad if someone sheds some light on the first section and the follow-up sections as I also started thinking about them in parallel.
It's preferable that we deal with each section one at time, since I'm sure there will be extra questions about each section independently.