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

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

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

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.