This book is based on the premise that engineers use probability as a modeling tool, and that probability can be applied to the solution of engineering problems. Engineers and students studying probability and random processes also need to analyze data, and thus need some knowledge of statistics. This book is designed to provide students with a thorough grounding in probability and stochastic processes, demonstrate their applicability to real-world problems, and introduce the basics of statistics. The book's clear writing style and homework problems make it ideal for the classroom or for self-study.
Chapter 2 Random Variables. Chapter 3 Moments of Random Variables.
Fundamentals of Applied Probability, Statistics, and Random Processes - 3rd Edition
Chapter 4 Special Probability Distributions. Chapter 5 Multiple Random Variables. Chapter 6 Functions of Random Variables. For instance, if the items are the four events A, B, C, D, the different arrangements possible, subject to the requirement that each arrangement must include exactly two of the items, are Such arrangements of N distinguishable items, taken K a t a time, are known as K-permutations of the items.
I n the above example, we have found by enumeration that there are exactly 12 different 2-permutations of 4 distinguishable items. To determine the number of K-permutations which may be formed from N distinguishable items, one may consider forming the permutations sequentially. One begins by choosing a first item, then a second item, etc; For instance, for the above example, this process may be shown on a sequential sample space, There were N choices for the first member, any such choice results in N - 1 possible choices for the second member, and this continues until there are N - K 1 possible choices for the Kth and last member of the permutation.
Two K-combinations are identical if they both select exactly the same' set of K items out of the original list and no attention is paid to the order of the items. For the example considered earlier, AB and BA are different permutations, b u t both are included in the same combination. To determine the number of K-combinations which may be formed from N distinguishable items,, we need only divide the number of K-permutations by K!
When one attacks such problems in an appropriate sequential event space, the problem is reduced to several simple counting problems,. The event-space approach provides the opportunity to deal only with the collection of mutually exclusive events in a collectively exhaustive space. This is a powerful technique and, to conclude this section, we note one simple example of its application.
I t is also specified that R and S cannot be on the same committee unless the committee contains at least one female. We first wish to determine the number of different such committees which may be formed. We draw an event space for the enumeration of these committees. Our only methodology is to decompose the counting problem into smaller pieces, always branching out into a mutually exclusive decomposition which includes all members of interest.
We may omit items which are not relevant to our interests. Event f,: Exactly n females are on the committee. For instance, for the RS' committees, note that there are as many acceptable committees containing R but not S as there are ways of selecting three additional members from T , U, V, W, X, Y, Z without regard to the order of selection. Discuss what evidence about the validity of the appropriate relation may be obtained from each of the following observations. We observe that we can: a Obtain a valid relation by taking the intersection of each side of relation 1 with some event El b Obtain an invalid relation by taking the intersection of each side of relation 2 with some event Ez c Obtain a valid relation by taking the union of each side of relation 3 with some event E3 d Obtain an invalid relation by taking the complement of each side of relation 4 1.
Had we decided to solve by counting unacceptable committees and subtracting from , we would have noticed immediately that only one draw, RST U, would be unacceptable. The extension of our event-space approach to problems involving permutations, such as ranked committees, merely requires that we work in a more fine-grained event space.
The die thus selected is then tossed until a white face appears, at which time the game is concluded. Die A has 4 red and 2 white faces. Die B has 2 red and 4 white faces. Determine the value of p. We use the notation - 1 p 3, I 1. If the dog is in A either dead or alive and Oscar spends a day searching for it in A, the conditional probability that he will find thedog that day is 0. Sinlilarly, if the dog is in B and Oscar spends a day looking for it there, he will find the dog that day with probability 0. The dog cannot go from one forest to the other. Oscar can search only in the daytime, and he can travel from one forest to the other only at night.
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All parts of this problem are to be worked separately. What is the a priori probability that he will find a live dog for the first timeon the second day? Given the fact that he was unsuccessful on the first day, determine the probability that he does not find a dead dog on the second day. He looked in A for the first 3 days and in B on the fourth day.
What is the probability he found his dog alive? The oaly other thing we know is that he looked in A for 2 days and and in B for 2 days. He quits as soon as he finds the dog. The Cardinals have only two pitchers, one of whom must start : C1, their best pitcher; C2, a worse pitcher.
The other team, the Yankees, has only three pitchers, one of whom must start: Yl, their best pitcher; Y2, a worse one; Yg, an even worse one. If C1 does start, the Yankees will start Y1 with probability 3. The Cardinals are equally likely to start C1 or C2, no matter what the Yankees do; Y t will pitch for the Yan1;ees with probability f if Y1 does not pitch, no matter what else occurs. Rlission success or failure depends only on the behavior of three major systems. Joe decides the following assumptions are valid and apply to the performance of an entire mission: 1 The mission is a failure only if two or more of the major systems fail.
If no other system fails, the Frab system will fail with probability 0. Otherwise, the Beer Cooler cannot fail. You are riot positive whnt the announcer did say, but, based on all available information, you decide thnt it is twice as likely that he reported "ALission a success'? A1 draws two coins from the box, flips each of then1 once, observes an outconle of one head and one tail, and returns the coins to the box. Do then draws one coin from the box and flips it. The result is a tail.
Determine the probability that neither A1 nor Bo removed the biased coin from the box. Joe, the bookie, is attempting to establish the odds on an exhibition baseball game. From many years of experience, he has learned that his prime colisideration should be the selection of the starting pitcher.
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If it falls heads, the game coritinues by throwing die A alone; if it falls tails, die B alone is used to continue t. However awful their face colors may be, it is known that both dice are fair. Interpret your result for large values of n. Events A and B are mutually exclusive and collectively exhaustive.
Events A and C are independent. Can C arid B be mutually exclusive? Towns A and B can communicate as long as they are connected in the communication network by at least one path which contains only in-service links. Determine, in an efficient manner, the probability that A and B can communicate.
Consider the physical situation before you write anything. A can communicate with B as long as they are connected by at least one path which contains only in-service links.
Bo writes inane jingles rapidly but poorly. He has probability 0. If Ci has not already won the contest, Bo's entry will be declared the winner with probability 0.
Ci writes slowly, but he has a gift for this sort of thing. If Bo has not already won the contest by the time of Ci's entry, Ci will be declared the winner with probability 0.
Can you find a value of P for which "First entry wins" and "Second entry does not win" are independent events? For each of the conditions below, state whether or not each statement following it must necessarily be true if the condition is true, and give a reason for your answer.
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Each midnight he stays iri the same room with probability 0. We can't observe the mouse directly, so we use the reports of an unreliable observer, who each day reports correctly with probability 0. Event Sk n :Jlouse is i r i room 1; on day n. Would IVC then believe the observer on the second day?
A committee of four delegates, selected by lot, is formed. Determine the probability that: a Company A is not represented on the committee.