$X_1+X_2$ and $X_1-X_2 $ are i.i.d. $N(1,4)$. What is the distribution of $X = (X_1,X_2)^T$?

I know i.i.d. is an independent and identically distributed random variable but I don't know how to use it to solve this problem that contains a column vector.

  • $\begingroup$ What does this have to do with computer science? $\endgroup$ Apr 4 at 20:19
  • $\begingroup$ Are you familiar with multivariate normal distributions? $\endgroup$ Apr 4 at 20:27
  • $\begingroup$ Please don't roll back edits that others make to your question. See cs.stackexchange.com/help/editing. $\endgroup$
    – D.W.
    Apr 5 at 4:50

Let $Y = (Y_1,Y_2)$ be two i.i.d. $N(1,4)$ variables. Then $Y$ is a multivariate normal distribution, and so any affine transformation of $Y$ is a multivariate normal distribution. In particular, if we define $X_1 = \frac{Y_1 + Y_2}{2}$ and $X_2 = \frac{Y_1 - Y_2}{2}$ then $X = (X_1,X_2)$ is a multivariate normal distribution which satisfies $X_1 + X_2 = Y_1$ and $X_1 - X_2 = Y_2$. In other words, it is the random variable you're interested in.

It is routine to calculate $$ (X_1,X_2) \sim N\left(\begin{pmatrix} 1 & 0 \end{pmatrix}, \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}\right), $$ that is, $X_1 \sim N(1,2)$ and $X_2 \sim N(0,2)$, and $X_1,X_2$ are independent.


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