1. Question about problem statement (marginal distribution)

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 $X is N(\mu,\sigma^2)$ and $Y is N(\mu,\sigma^2)$ 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 $X_i is n(\mu_i, \sigma_i^2)$. Let $a_1, \ldots, a_n$ and $b_1, \ldots, b_n$ be fixed constants Then

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

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.

2. Re: Question about problem statement (marginal distribution)

Hey abscissa.

Do you mean finding the marginal distributions of X and Y given a joint distribution U = f(X,Y)? As you have pointed out since X and Y are independent then the joint distribution is P(X = x, Y = y) = P(X=x)*P(Y=y) and integrating out one will give the other.

Use the theorem to show that the sum of normals has the means summed and the variances summed as well.