3. A fair die is rolled twice, with outcomes X for the 1st roll and Y for the 2nd roll. (a) Compute the covariance of X + Y and X - Y (simplify). (b) Are X + Y and X - Y independent? Justify your answer clearly.
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Since X and Y are independent and represent the outcome of a fair die, we have: $$E[X] = E[Y] = \frac{1}{6}(1+2+3+4+5+6) = 3.5$$ $$E[X+Y] = E[X] + E[Y] = 3.5 + 3.5 = 7$$ $$E[X-Y] = E[X] - E[Y] = 3.5 - 3.5 = 0$$ Show more…
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A fair die is rolled twice independently with outcomes X for the first roll and Y for the second roll. Compute the covariance of X + Y and X - Y. Are X + Y and X - Y independent? Justify your answer clearly.
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