# How to compute the correlation coefficient?

The question is:

One package of potatoes contains 10 potatoes and weighs exactly 500 grams. Denote by $$X_{1}, \dots, X_{10}$$ the weights of each potato.

Are the random variables $$X_{1}, \dots, X_{10}$$ independent?

Compute the correlation coefficient of $$\rho(X, Y)$$ where $$X=X_{1}$$ and $$Y = \sum_{i=2}^{10} X_{i}$$

I know this formula $$\rho=\frac{cov(X,Y)}{\sigma_{X} \sigma_{Y}}$$ and that $$cov(X,Y)=E[XY] - E[X]E[Y]$$

So it seems that it is just to plug in the right values and compute. But Im not sure how to calculate $$E[X]$$ and $$E[Y]$$..

I think it is something along with: I know that $$E[X]=xf(x)$$ and here $$x=X_{1}$$ and $$f(x) = 1$$ soo this equals $$X_{1}$$? This is true (since this set only contains this potato so therefore we must always get it when we choose). But the answer should be a number, not a random variable...

The same goes for $$E[Y]$$.

I know from the solutions that the answer is: $$\rho(X,Y)=-1$$ and thus they are in correlation.

Define:

$$\sum_{i = 2}^{10}X_{i} = Y$$

We know that:

$$500 = X_{1} + Y \Rightarrow Y = 500 - X_{1}$$

from which:

$$Cov(X_{1}, Y) = Cov(X_{1}, 500 - X_{1}) = Cov(X_{1}, -X_{1}) = -Var(X_{1})$$

using the follwing covariance properties:

$$Cov(a + X, b + Y) = Cov(X, Y); \ Cov(aX, bY) = abCov(X, Y); \ Cov(X,X) = Var(X)$$

Moreover:

$$Var(Y) = Var(500 - X_{1}) = Var(X_{1})$$

also using $$Var(a + X)$$ $$=$$ $$Var(X)$$ and $$Var(-X)$$ = $$Var(X)$$, from which we finally have:

$$\sqrt{Var(X_{1})}\sqrt{Var(Y)} = \sqrt{Var(X_{1})}\sqrt{Var(X_{1})} = Var(X_{1})$$

implying $$\rho(X_{1}, Y)$$ $$=$$ $$-1$$