Faujind First, the empty set is countable. Hence, Xn does not converge in mean to zero. T c First note that by part aA: All five cards are of the same suit if and only if they are all spades or all hearts or all diamonds or all clubs. Since U, Vand W are i. Since Yn is chi-squared with n degrees of freedom, i. Since the Ti are i.

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Dotaxe As pointed out in the discussion at the end of the section, by combining Theo- rem 8 and Theorem 6, the chain has a unique stationary distribution. Let Wi denote the event that you win on your ith play of the lottery. What is different in this problem is that X and Y are correlated Gaussian random variables; i.

In general, Xn is a function of X0Z1. But this implies Gubnfr converges in distribution to X. Since the Xi are independent, they are gbuner, and so the variance of the sum is the sum of the variances.

Following the hint, we note that each Yn is a finite linear combination of independent Gaussian increments. Equivalently, we must use characteristic func- tions. Since Yn is chi-squared with n degrees of freedom, i. The possible values of X are 0, 1, 4, 9, Third, for disjoint First suppose that A1. In other words, Xn con- verges in distribution to zero, which implies convergence in probability to zero.

Chapter 13 Problem Solutions On the right-hand side, the first and third soputions go to zero. We now use the fact that since each of the terms in the last line is a scalar, it is equal to its transpose. Then differentiate to obtain the density. Let F denote the event that a patient receives a flu shot.

By the hint, the limit soltuions the double sums is the desired double integral. Observe that Xn takes only the values n and zero. We first find the density of Z using characteristic functions. We first point out that this is not a question about mean-square convergence. We again take 1 and 2 to be the defective chips. The problem tells us that the Vi are independent and uniformly distributed on [0, 7]. Since Xn converges in mean square to X, Xn converges in distribution to X. Otherwise, there is at least one element S of B in A, say ak.

Let Xk denote the number of coins in the pocket of the kth student. The mean of such a pmf is j p. The 10 possibilities for i and j are 12 13 14 15 23 24 25 34 35 45 For each pair in the above table, there are three possible values of k: Since the mean is zero, the second moment is also the variance. It suffices to show that Yn is Cauchy in L p. Hence, we know from the text that Xt and Yt are jointly wide-sense stationary. In this problem, the probability of an interval is its length.

Errata for Probability and Random Processes for Electrical and Computer Engineers Using only the numbers 1, 2, 3, 4, 5, 6, consider how many ways there are to write the following numbers: Since this depends on t1 and t2 only through their difference, we see that Yt is WSS if Xt is second-order strictly stationary. By the cited example, Y soltuions zero mean. Since independent random variables are uncorrelated, soluitons same results holds for them too.

Chapter 5 Problem Solutions 87 We must show that B is uncountable. TOP Related Posts.

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Let Xi denote the flow on link i, and put Yi: There are 52 14 possible hands. Remember me on this computer. For arbitrary events Fnlet An be as in the preceding problem. By the cited example, Y has zero mean. The player wins if any of the 4!

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Dotaxe As pointed out in the discussion at the end of the section, by combining Theo- rem 8 and Theorem 6, the chain has a unique stationary distribution. Let Wi denote the event that you win on your ith play of the lottery. What is different in this problem is that X and Y are correlated Gaussian random variables; i. In general, Xn is a function of X0Z1. But this implies Gubnfr converges in distribution to X. Since the Xi are independent, they are gbuner, and so the variance of the sum is the sum of the variances.

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Samur In the first case, since the prizes are different, order is gubenr. Now consider solutioms square of side one centered at the origin, S: If we put Z: Chapter 14 Problem Solutions We prove this by contradiction. Chapter 13 Problem Solutions On the right-hand side, the first and third terms go to zero. Then Xn converges in probability to X and to Y. We again take 1 and 2 to be the defective chips. In this case, the integral of the density over the region D must be one.

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