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As underlined in red, please help me with problem 2 (preferably following the hint) and 3 ii)

Thank

you!

416HW4.jpg

416HW4.jpg

2. There are n couples (A1, B1), ..., ( An, Bn). Suppose there are to be seated on a row
of 2n chairs randomly. Let X be the total number of couples who sit next to their
spouse. Find the expectation E(X ) and Var(X). Hint: For each couple , define
some random variables Xi which takes either one or zero, depending on ... When
calculating the variance, note that X; and Xj are dependent for i # j. Thus, we
need to compute the covariances Cov(Xi, Xi).
3. (Exercise # 68 in Chap.2, with some modification) Let X1, ..., X10 be independent
and identically distributed random variables with mean 1 and variance 1.
(i) Use the Markov inequality to get a bound on P{Xi + . . . + X10 > 15}.
(ii) Use the central limit theorem to approximate the above probability.

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