I am doing some problems from a practice final and would like to know if the following problem has mistakes in the way it is written. We are supposed to apply a corollary that doesn't seem to have any relevance in this context. It is throwing me off.

Problem statement: Suppose that $\displaystyle $X$ is $N(\mu,\sigma^2)$$ and $\displaystyle $Y$ is $N(\mu,\sigma^2)$$ and they are independent. Let $\displaystyle $U=X+Y$$ and $\displaystyle $V=X+Y$$. Use the following corollary to find the marginal distributions of $X$ and $Y$.

Corollary: Let $\displaystyle $X_1, \ldots, X_n$$ be mutually independent random variables with $\displaystyle $X_i$ is $n(\mu_i, \sigma_i^2)$$. Let $\displaystyle $a_1, \ldots, a_n$$ and $\displaystyle $b_1, \ldots, b_n$$ be fixed constants Then

$\displaystyle $Z=\sum_{i=1}^n(a_iX_i + b_i)$ is $n(\sum_{i=1}^n(a_i\mu_i + b_i),\sum_{i=1}^na_i^2\sigma_i^2)$$.

Also, aren't the marginal distributions of $X$ and $Y$ just $X$ and $Y$ themselves, because they are independent of each other??

Any help would be greatly appreciated. My final is tomorrow and I'm studying as hard as I can.