Write X = ∑i=1^n Xi and Y = ∑j=1^n Yj where Xi = 1 if the i-th roll is a 1 and zero otherwise. S

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Write $X = \sum_{i=1}^{n} X_i$ and $Y = \sum_{j=1}^{n} Y_j$ where $X_i = 1$ if the i-th roll is a 1 and zero otherwise. Similarly, $Y_j = 1$ if the j-th roll is a 2 and zero otherwise. Then $cov(X, Y) = \sum_{i,j} cov(X_i, Y_j)$. Next compute $cov(X_i, Y_j) = E(X_i Y_j) - E(X_i)E(Y_j)$. Note that $E(X_i Y_j)$ is non-zero only when $i \neq j$.

          Write $X = \sum_{i=1}^{n} X_i$ and $Y = \sum_{j=1}^{n} Y_j$ where $X_i = 1$ if the i-th roll is a 1 and zero otherwise. Similarly, $Y_j = 1$ if the j-th roll is a 2 and zero otherwise.
Then $cov(X, Y) = \sum_{i,j} cov(X_i, Y_j)$. Next compute
$cov(X_i, Y_j) = E(X_i Y_j) - E(X_i)E(Y_j)$. Note that $E(X_i Y_j)$ is non-zero only when $i \neq j$.

Write X = ∑i=1^n Xi and Y = ∑j=1^n Yj where Xi = 1 if the i-th roll is a 1 and zero otherwise. Similarly, Yj = 1 if the j-th roll is a 2 and zero otherwise.
Then cov(X, Y) = ∑i,j cov(Xi, Yj). Next compute
cov(Xi, Yj) = E(Xi Yj) - E(Xi)E(Yj). Note that E(Xi Yj) is non-zero only when i ≠ j.

Added by Raul B.

Elementary Statistics a Step by Step Approach

Allan G. Bluman 9th Edition

Instant Answer

Solved by Expert Adi S

09/20/2022

Step 1

Step 1: We know that $cov(X, Y) = \sum_{i,j} cov(X_i, Y_j)$. Show more…

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Write $X = \sum_{i=1}^n X_i$ and $Y = \sum_{j=1}^n Y_j$ where $X_i = 1$ if the i-th roll is a 1 and zero otherwise. Similarly, $Y_j = 1$ if the j-th roll is a 2 and zero otherwise. Then $cov(X, Y) = \sum_{i,j} cov(X_i, Y_j)$. Next compute $cov(X_i, Y_j) = E(X_iY_j) - E(X_i)E(Y_j)$. Note that $E(X_iY_j)$ is non-zero only when $i \neq j$.