Distribution of vector of normal random variables

Question

Suppose that $X_1, X_2, X_3$ are i.i.d. normal random variables with mean $\mu=0$ and variance $\sigma^2=1$. $X_i\sim N(0, 1)$ for $i=1,2,3$.

(a) Define a vector $X=(X_1,X_2,X_3)$. What is the distribution of $X$? I'm not sure how to answer this. $X$ is just a vector with three components that each have the distribution $N(0, 1)$. Do I just put that?

(b) Let $Y=(X_1, X_1+X_2)$. What is the distribution of $Y$? Here $X_1$ appears in two components, so they are linked. How do I address that?

(c) Suppose $Z\sim N(1, 4)$ and is independent of the $X_i,i=1,2,3$. What is the distribution of the vector $(Z+X_1, Z+X_2, Z+X_3)$? What is the correlation coefficient between $Z+X_1$ and $Z+X_2$?

Answer 1

I assume that when you are asked to provide the distribution, that it implies the distribution function.

Well, for the first question. You can easily find $F_{X_1,X_2,X_3}(x_1, x_2,x_3)$ using their independence; it factorizes into the product of three individual distribution functions.

For the second one, write $X_1 + X_2 = Z$. So we look for $F_{X_1,Z}(x_1,z)$, which is defined to be: $F_{X_1,Z}(x_1,z)=\mathbb{P}(X_1\leq x_1, Z\leq z)=\mathbb{P}(X_1\leq x_1, X_1 + X_2\leq z)$. I think I ought to better keep it to this, as I believe you are able to proceed yourself from this point.