$\{ X_{i} \}_{i=1}^{n} \thicksim iid N(\theta, 1)$. What is distribution of $X_{2} - X_{1}$?

Question

So the full question is:

Let $\{ X_{i} \}_{i=1}^{n} \thicksim \text{iid N}(\theta, 1)$. Show then that $\text{T}(X)=X_{2}-X{1} \thicksim \text{N}(0,2)$.

Now, I get that $\mathbb{E}[T(X)]=0$. But I do not get why the variance goes from 1 to 2? Can someone please explain? I didn't think the variance would change? ie, in my view it should be $\mathbb{Var}[\text{T}(X)] = 1$.

For your reference, this is to do with sufficient statistics and ancillatry (ancillary statistics).

Answer 1

$\mathbb{Var}(aX_1+bX_2)=a^2\mathbb{Var}(X_1)+b^2\mathbb{Var}(X_2)$ if $X_1$ and $X_2$ are independent, so $\mathbb{Var}(X_1-X_2)=\mathbb{Var}(1\cdot X_1+(-1)\cdot X_2)=1^2\mathbb{Var}(X_1)+(-1)^2\mathbb{Var}(X_2)=\mathbb{Var}(X_1)+\mathbb{Var}(X_2)=1+1=2$.