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Bernoulli p random variables is a binomial n, p. Gubndr that, the Chernoff bound is the smallest. The possible values of X are 0, 1, 4, 9, We know from our earlier work that the Wiener integralR is linear on piecewise-con- R stant functions.

For this choice of pn Xn converges almost surely but not in mean to X.

Suppose Xn gubnrr almost surely to X, and Xn converges in mean to Y. In other words, as a function of i, pX Z i j is a binomial j, p pmf.


These are four disjoint events. This is an instance of Problem All five cards are of the same suit gubnrr and only if they are all spades or all hearts or all diamonds or all clubs. Since this depends on t1 and t2 only through their difference, we see that Yt is WSS if Xt is fourth-order strictly stationary.

Next, since the Xk are i. The true probability is 0.

Errata for Probability and Random Processes for Electrical and Computer Engineers

For such s both integrals are easy to evaluate. The left-hand side is more work: Hence, each Yn is Gaussian. First note that since X and W are zero mean, so is Y. Denote the arrival times of Nt by T1 bubner, T2.

We show this to be the case. Third, for disjoint There are k1By the cited example, Y has zero mean. The remaining sum gubnrr obviously nonnegative.

Frame ALERT!

Since the Ti are i. By Problem 11, V 2 is chi-squared with one degree of freedom. By the hint, the limit gunner the double sums is the desired double integral. The second term on the right is equal to zero because T is linear on trigonometric polynomials. Now consider the square of side one centered at the origin, S: Hence, Wt converges in distribution to the zero random variable.


Bubner particular, this means R that the left-hand side integrates to one. Then the time to transmit n packets is T: For any constants c1.

Let Xk denote the number of coins in the pocket of the kth student. Suppose A is countable and B is uncountable.

Following the hint, we put Gn: Chapter 3 Problem Solutions 41 It will then follow that the increments are independent. By the hint, [Wt1 .