The original question is:
For $t\in [0,1]$, we define $X_t=B_t-tB_1$, where $\{B_t:t\geq 0\}$ is a standard Brownian motion. Find the density of $X_t$ .
After reading several resources, I think $X_t$ is a normal distribution. However, since most of the books concerning on the process itself other than the distribution of $X_t$, can someone confirm this?