Random vector and the correlation

Question

You got a random vector $X = (x, y)$, the random variables $x$ and $y$ consist of $40$ objects.

$x \sim \text{normal distribution}(23, 48)$

$y = 23x + U$

$U \sim \text{normal distribution}(0, 29)$

You need to find the correlation coefficent. What should I start doing? We need to do it inside excel, but for correlation dont I need some points and so on? Right now i just got definitions of the random variables, how should i proceed?

Pardon my english

Edit:

the correlation is for (x, y). What I did try was make a table in excel with integers from 0 to 40 and then for x, this is what I used to get the table:

=NORM.DIST(i;23;48;TRUE), where i is the current integer. Then from that I just made the y-s as well following similar logic with U, ended up with a correlation coefficent of 1 which I dont think is right and I think i made something wrong

Answer 1

\begin{align} \operatorname{Cov}(X,Y)&=\mathbb{E}[(X-\mathbb{E}[X])(Y-\mathbb{E}[Y])]\\ &=\mathbb{E}[(X-\mathbb{E}[X])((23X+U)-\mathbb{E}[23X+U])]\\ &=\mathbb{E}[(X-\mathbb{E}[X])(23(X-\mathbb{E}[X]+(U-\mathbb{E}[U])]\\ &=23\mathbb{E}[(X-\mathbb{E}[X])(X-\mathbb{E}[X])] +\mathbb{E}[(X-\mathbb{E}[X])(U-\mathbb{E}[U])]\\ &=23\operatorname{Var}(X)+\operatorname{Cov}(X,U) \end{align}

So you just substitute the definition of $Y$ into the expression and use linearity of the expectation value and basic algebraic transformations to get to the last expression. If you actually want the correlation you will have to divide by the standard deviation of Y and X. Calculating the Variance of Y will be similar to the Covariance - again plug in the definition and transform it until it is an expression of known things such as the variance of $X$ or $U$.

And if you are just supposed to simulate a bunch of those variables and calculate the empirical values - well do that. But I don't think you would need excel otherwise.

Answer 2

$\newcommand{\cov}{\operatorname{cov}}\newcommand{\var}{\operatorname{var}}$ \begin{align} \cov(x,y) & = \cov(x,23x+U) \\[10pt] & = \cov(x,23x) + \cov(x,U) \\[10pt] & = 23\cov(x,x) + 0 \\ & \qquad\text{if }x,U \text{ are independent (but you withheld that information from us).} \\[10pt] & = 23\var(x). \\[10pt] \var y & = \var(23x+U) \\[10pt] & = \var(23x) + \var U + 2\cov(23x,U) \\[10pt] & = 23^2 \var x + \var U \\ & \qquad \text{(again, if independence holds).} \\[10pt] \operatorname{correlation}(x,y) & = \frac{\cov(x,y)}{\sqrt{\var x}\sqrt{\var y}}. \end{align}