Discrete Probability Distributions 1
In this book we shall study many different experiments from a probabilistic point of view. What is involved in this study will become evident as the theory is developed and examples are analyzed. However, the overall idea can be described and illustrated as follows to each experiment that we consider there will be associated a random variable, which represents the outcome of any particular experiment. The set of possible outcomes is called the sample space. In the first part of this section, we...
Info Gjn
We now turn to the question of what happens when we riffle shuffle s times. It should be clear that if we start with the identity ordering, we obtain an ordering with at most 2s rising sequences, since a riffle shuffle creates at most two rising sequences from every rising sequence in the starting ordering. In fact, it is not hard to see that each such ordering is the result of s riffle shuffles. The question becomes, then, in how many ways can an ordering with r rising sequences come about by...
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In the above coin tossing and the dice rolling experiments, we have assigned an equal probability to each outcome. That is, in each example, we have chosen the uniform distribution function. These are the natural choices provided the coin is a fair one and the dice are not loaded. However, the decision as to which distribution function to select to describe an experiment is not a part of the basic mathematical theory of probability. The latter begins only when the sample space and the...
Random Variables and Sample Spaces
Definition 1.1 Suppose we have an experiment whose outcome depends on chance. We represent the outcome of the experiment by a capital Roman letter, such as X, called a random variable. The sample space of the experiment is the set of all possible outcomes. If the sample space is either finite or countably infinite, the random variable is said to be discrete. We generally denote a sample space by the capital Greek letter Q. As stated above, in the correspondence between an experiment and the...
Tree Diagrams
Example 1.10 Let us illustrate the properties of probabilities of events in terms of three tosses of a coin. When we have an experiment which takes place in stages such as this, we often find it convenient to represent the outcomes by a tree diagram as shown in Figure 1.8. A path through the tree corresponds to a possible outcome of the experiment. For the case of three tosses of a coin, we have eight paths wi, w2, , w8 and, assuming each outcome to be equally likely, we assign equal weight, 1...
Random Numbers
We must first find a computer analog of rolling a die. This is done on the computer by means of a random number generator. Depending upon the particular software package, the computer can be asked for a real number between 0 and 1, or an integer in a given set of consecutive integers. In the first case, the real numbers are chosen in such a way that the probability that the number lies in any particular subinterval of this unit interval is equal to the length of the subinterval. In the second...
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Figure 4.2 Reverse tree diagram. Bayes Probabilities Our original tree measure gave us the probabilities for drawing a ball of a given color, given the urn chosen. We have just calculated the inverse probability that a particular urn was chosen, given the color of the ball. Such an inverse probability is called a Bayes probability and may be obtained by a formula that we shall develop later. Bayes probabilities can also be obtained by simply constructing the tree measure for the two-stage...
Exercises
1 Modify the program CoinTosses to toss a coin n times and print out after every 100 tosses the proportion of heads minus 1 2. Do these numbers appear to approach 0 as n increases Modify the program again to print out, every 100 times, both of the following quantities the proportion of heads minus 1 2, and the number of heads minus half the number of tosses. Do these numbers appear to approach 0 as n increases 2 Modify the program CoinTosses so that it tosses a coin n times and records whether...
Card Shuffling
Much of this section is based upon an article by Brad Mann,28 which is an exposition of an article by David Bayer and Persi Diaconis.29 Given a deck of n cards, how many times must we shuffle it to make it random Of course, the answer depends upon the method of shuffling which is used and what we mean by random. We shall begin the study of this question by considering a standard model for the riffle shuffle. We begin with a deck of n cards, which we will assume are labelled in increasing order...
Combinatorics
Many problems in probability theory require that we count the number of ways that a particular event can occur. For this, we study the topics of permutations and combinations. We consider permutations in this section and combinations in the next section. Before discussing permutations, it is useful to introduce a general counting technique that will enable us to solve a variety of counting problems, including the problem of counting the number of possible permutations of n objects. Consider an...
Continuous Conditional Probability
In situations where the sample space is continuous we will follow the same procedure as in the previous section. Thus, for example, if X is a continuous random variable with density function f x , and if E is an event with positive probability, we define a conditional density function by the formula f x E 0, if x E. Then for any event F, we have The expression P F E is called the conditional probability of F given E. As in the previous section, it is easy to obtain an alternative expression for...
PHi n E P E
We can calculate the numerator from our given information by P Hi n e P Hi P E Hi . 4.2 Since one and only one of the events hi, h2, , Hm can occur, we can write the probability of E as P e P Hi n e P h2 n e P Hm n E . Using Equation 4.2, the above expression can be seen to equal P hi p e hi P H P e h2 P Hm P E Hm . 4.3 Using 4.1 , 4.2 , and 4.3 yields Bayes' formula Although this is a very famous formula, we will rarely use it. If the number of hypotheses is small, a simple tree measure...
Exercises Yyz
1 In the spinner problem see Example 2.1 divide the unit circumference into three arcs of length 1 2, 1 3, and 1 6. Write a program to simulate the spinner experiment 1000 times and print out what fraction of the outcomes fall in each of the three arcs. Now plot a bar graph whose bars have width 1 2, 1 3, and 1 6, and areas equal to the corresponding fractions as determined by your simulation. Show that the heights of the bars are all nearly the same. 2 Do the same as in Exercise 1, but divide...
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Now we assume that when the needle drops, the pair 0, d is chosen at random from the rectangle 0 lt 0 lt n 2, 0 lt d lt 1 2. We observe whether the needle lies across the nearest line i.e., whether d lt 1 2 sin 0 . The probability of this event E is the fraction of the area of the rectangle which lies inside E see Figure 2.5 . Figure 2.5 Set E of pairs 9, d with d lt 1 sin 9. Now the area of the rectangle is n 4, while the area of E is The program BuffonsNeedle simulates this experiment. In...
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Hint Consider an urn with n red balls and n blue balls inside. Show that each side of the equation equals the number of ways to choose n balls from the urn. 36 Let j and n be positive integers, with j lt n. An experiment consists of choosing, at random, a j-tuple of positive integers whose sum is at most n. a Find the size of the sample space. Hint Consider n indistinguishable balls placed in a row. Place j markers between consecutive pairs of balls, with no two markers between the same pair of...
Charles Claims That He Can Distinguish Between Beer And Ale
2 In how many ways can we choose five people from a group of ten to form a committee 3 How many seven-element subsets are there in a set of nine elements 4 Using the relation Equation 3.1 write a program to compute Pascal's triangle, putting the results in a matrix. Have your program print the triangle for n 10. 24A. W. F. Edwards, op. cit., p. ix. 25 J. Bernoulli, Ars Conjectandi Basil Thurnisiorum, 1713 . 5 Use the program BinomialProbabilities to find the probability that, in 100 tosses of a...
Paradoxes
Much of this section is based on an article by Snell and Vanderbei.18 One must be very careful in dealing with problems involving conditional probability. The reader will recall that in the Monty Hall problem Example 4.6 , if the contestant chooses the door with the car behind it, then Monty has a choice of doors to open. We made an assumption that in this case, he will choose each door with probability 1 2. We then noted that if this assumption is changed, the answer to the original question...
Suppose You Are Watching A Radioactive Source That Emits Particles At A Rate
We note that it is not the case that all continuous real-valued random variables possess density functions. However, in this book, we will only consider continuous random variables for which density functions exist. In terms of the density f x , if E is a subset of R, then The notation here assumes that E is a subset of R for which fE f x dx makes sense. Example 2.10 Example 2.7 continued In the spinner experiment, we choose for our set of outcomes the interval 0 lt x lt 1, and for our density...
Explain Why It Is Not Possible To Define A Uniform Distribution Function On A
19 If A, B, and C are any three events, show that - P A n b - P B n C - P C n A P a n B n C . 20 Explain why it is not possible to define a uniform distribution function see Definition 1.3 on a countably infinite sample space. Hint Assume m w a for all w, where 0 lt a lt 1. Does m w have all the properties of a distribution function 21 In Example 1.13 find the probability that the coin turns up heads for the first time on the tenth, eleventh, or twelfth toss. 22 A die is rolled until the first...