I have $2$ independent random variables - $X$ and $Y$: $$ X \sim \operatorname{N}\left(0,\sigma_{x}^{2}\right),\quad Y \sim \operatorname{N}\left(\mu, \sigma_{y}^{2}\right) $$
Define $Z = X + Y$
$5$ observations ( or in general $n$ ): $\left(x_{1},y_{1}\right),\ldots,\left(x_{5},y_{5}\right)$
My question is - How do I formulate the conditional distribution of $x_{1}$, given that I know $z_{1}$ is the maximum of $z_{1},z_{2},\ldots,z_{5}\ ?$.
( I know the distribution of $Z$, since it is the sum of $2$ independent normals, and I know the distribution of $\operatorname{max}\left(z\right)$ - which is given by the order statistics - but I am unable to proceed to get the conditional distribution as described ).
(Edit: If it makes it any simpler, can assume $X$ and $Y$ to be uniformly distributed instead of normal).
Edit2: If 5 makes it tricky, what if we consider the simpler case of just 2 observations, $z_{1},z_{2}$ ?
Any help would be appreciated, thanks!.
By the Law of Total Probability:
$\qquad\begin{align}f_{[X_1\mid Z_1=\max\{Z_k\}]}(x)&=\int_\Bbb R f_{[X_1,Z_1\mid Z_1=\max\{Z_k\}]}(x,z)\,\mathrm d z\\[2ex]&=\int_\Bbb R\dfrac{f_{X}(x)\,f_{Y}(z-x)\,{F_{[X+Y]}}^{n-1}(z)}{\mathsf P(Z_1=\max\{Z_k\})}\,\mathrm d z\\[2ex]&= n\, f_{X}(x)\int_\Bbb R f_{Y}(z-x)\,{F_{[X+Y]}}^{n-1}(z)\,\mathrm d z\\[3ex]&= 5\, f_{X}(x)\int_\Bbb R f_{Y}(z-x)\,{F_{[X+Y]}}^{4}(z)\,\mathrm d z&~\ni~&n=5\end{align}$