Solution
= Solution
The pair $(X_1,T)$ is jointly normal, with
$$
\mathbb EX_1=\mu,\quad \mathbb ET=2\mu,\quad
\operatorname{var}(X_1)=1,\quad
\operatorname{var}(T)=2,\quad
\operatorname{cov}(X_1,T)=1.
$$
The <conditional distribution of a bivariate normal variable> is therefore
$$
X_1\mid T=t\sim N\left(\frac t2,\frac12\right).
$$
Equivalently, its conditional density is
$$
\boxed{f_{X_1\mid T=t}(x)=\frac1{\sqrt\pi}
\exp\left[-\left(x-\frac t2\right)^2\right].}
$$