John A. Gubner received his PhD in from the University of Maryland, College Park, after which he joined the University of Wisconsin, Madison, where he is currently a faculty member in the Department of Electrical and Computer Engineering. His research interests include ultra-wideband communications, point processes and shot noise, subspace methods in statistical processing, and information theory. The theory of probability is a powerful tool that helps electrical and computer engineers to explain, model, analyze, and design the technology they develop. The text begins at the advanced undergraduate level, assuming only a modest knowledge of probability, and progresses through more complex topics mastered at graduate level.

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Generally, an eBook can be downloaded in five minutes or less Show More. Han Nguyen. Ahmed Ganz , No Downloads. Views Total views. Actions Shares. Embeds 0 No embeds. No notes for slide. Chapter 1 Problem Solutions 3 8. We must show that C A. But then w 2 A B [C , which implies w 2 A. Therefore, w 2 B, and then w 2 AB.

We must show that w 2 I too. In other words, we must show that w 2 A w 2 B. But we already have w 2 B. Y invertible. Let f :X! Y be a function such that f takes only n distinct values, say y1,. But f x must be one of the values y1,. We must show that B is countable. Otherwise, there is at least one element S of B in A, say ak. Let A B where A is uncountable. We must show that B is uncountable. We prove this by contradiction. Suppose that B is countable.

Then by the previous problem, A is countable, contradicting the assumption that A is uncountable. Suppose A is countable and B is uncountable. Wemust show that A[B is uncountable. Suppose that A [ B is countable. Then since B A [ B, we would have B countable as well, contradicting the assumption that B is uncountable.

We must check the four axioms of a probability measure. First, P? Third, for disjoint An, suppose w0 2 S n An. Chapter 1 Problem Solutions 7 The hard part is to show the reverse inclusion.

Then w 2 Fn for some n in the range 1,. However, w may belong to Fn for several values of n since the Fn may not be disjoint. For arbitrary events Fn, let An be as in the preceding problem. Now suppose the union bound holds for some N 2. To establish the union bound for a countable sequence of events, we proceed as fol-lows.

In this problem, the probability of an interval is its length. We first show that this collection is a s-field. First, it contains? Second, since A1,. Hence, the collection is closed under complementation. This shows that the collection is a s-field.

Finally, since every element in our collec-tion must be contained in every s-field that contains A1,. We claim that A is not a s-field. Our proof is by contradiction: We assume A is a s-field and derive a contradiction. Now, any set in A must belong to some An. By the Suppose that a s-field A contains an infinite sequence Fn of sets. If the sequence is not disjoint, we can construct a new sequence An that is disjoint with each An 2 A.

Furthermore, since the Ai are disjoint, each sequence a gives a different union, and we know from the text that the number of infinite sequences a is uncountably infinite.

Hence, Ac 2 T a Aa. Third, if An 2 A for all n, then for each n and each a, An 2 Aa. Finally, if D is any s-field that contains C , then D is one of the A s in the intersection. Chapter 1 Problem Solutions 11 The union of two s-fields is not always a s-field. Here is an example. Let W denote the positive integers, and let A denote the collection of subsets A such that either A or Ac is finite.

Then E does not belong to A since neither E nor Ec the odd integers is a finite set. First suppose that A1,. Then In the second case, suppose that some Acj is finite. Let W be an uncountable set. Let A denote the collection of all subsets A such that either A is countable or Ac is countable. We show that A is a s-field.

First, the empty set is countable. Second, if A 2 A , we must show that Ac 2 A. There are two cases. If Ac is countable, then Ac 2 A. Third, let A1,S A2,. There are two cases to consider. If all An are countable, then n An is also countable by an earlier problem. Let I denote the collection of open intervals, and let O denote the collection of open sets.

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## Probability and Random Processes for Electrical and Computer Engineers

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