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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.

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

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