Rules for Means and Variances; PredictionRules for Means and VariancesPredicting for Bernoulli TrialsWhen p is KnownWhen p is UnknownPredicting for a Poisson ProcessSummaryPractice ProblemsSolutions to Practice ProblemsHomework ProblemsChapter 14Rules for Means and Variances; Prediction14.1 Rules for Means and VariancesThe result in this section is very technical and algebraic. And dry. But it is useful for understandingmany of prediction results we obtain in this course, beginning later i n this chapter.We have independent random variables W1and W2. Note that, typi call y, they are not identi call ydistributed. The result below is an extremely special case of a much more general result. It wi llsuffice, however, for our needs; thus, I see no reason to subject you to the pain of viewing thegeneral result.We need some notation:• Let µ1[µ2] denote the mean of W1[W2].• Let Var(W1) [Var(W2)] denote the variance of W1[W2].• Let b denote any number. DefineW = W1− bW2.Result 14.1 (The mean and variance of W .) For t he notation given above,• The mean of W isµW= µ1− bµ2(14.1)• The variance of W isVar(W ) = Var(W1) + b2Var(W2). (14.2)In our two applications o f t h is result in thi s chapter, the number b will be taken to equal µ1/µ2;thus,µW= µ1− (µ1/µ2)µ2= µ1− µ1= 0,for our applications.33714.2 Predicting for Bernoulli TrialsPredictions are tough, especially about the future—Yogi Berra.We plan to obs erve m Bernoulli trials and want to predict the to tal n umber of su ccesses that willbe obtained. Let Y denote the random variable and y the observed value of the total numberof successes in the future m trials. Similar to estimation, we will learn about point and intervalpredictions of the value of Y .14.2.1 When p is KnownBecause prediction is a new idea in this course, I want to present a gentle introduction to it. Supposethat y ou have a favorite pair of dice, one colored blue and the other white. Let’s focus on the bluedie. A nd let’s say your favorite number is 6—perhaps you play Risk a great deal; in Risk, 6 is thebest outcome by far when one casts a die. Ghengis Khan playing Risk would roll a lot of 6’s.You plan to cast the die 600 times and you want to predict the number of 6’s that you willobtain. You believe that the die is balanced; i.e., t hat the six possible outcomes are equally likelyto occur.OK. Quick. Don’t think of any of t he wonderful t hings yo u have learned in this course. Giveme your point (single number) prediction of how many 6’s you will obtain. I conjecture that youranswer is 100. (I asked this question several times over the years to a live lecture and always—saveonce—received the answer 100 from the student who volunteered to answer. One year a guy said72 and got a big laugh. I failed him because it is my job to make the jokes, such as they are. No, Ididn’t really fail him, but I was more than a bit annoyed that he got a larger laugh than I did withmy much cleverer anecdotes.)My academic grandfather (my advisor’s advi sor, who happened to be male) is Herb Robb ins,a very brilliant and witty man. Herb was once asked what mathematical stat isticians do, and hereplied, “They find out what non-statisticians do and prove it’s optimal.”Thus, I am going to argue th at your answer of 1 0 0 is the best answer to the die question I posedabove. In order to show that something is best mathemati call y, we find a way to measure goodand whichever candidate answer has the largest amount of good is best. This is the approach forthe glass-half-full people. More o ft en, one finds a way to measure bad and whichever candidateanswer has the smallest amount of bad is best.We want to predict, in advance, the value that Y will yield. We denote the point prediction bythe single number ˆy. We adopt the criterion th at we want the probabi lity of being correct to be aslarge as poss ible. (Thus, we d efine being correct as good and seek to maximize the probability thatwe will get a good result.)Result 14.2 (The best point prediction of Y .) Calculate the mean o f Y , which is mp.• If mp is an integer, then it is uniquely the most probable value of Y and our poi nt predictionis ˆy = mp.338• If mp is not an integer, then the most pro b able value of Y is one of the integers immediatelyon either side of mp. Check them both; whichever is more probab le is the point prediction.If they are equally probably, I choose the even integer.Below are some examples of this result.• For my die example, m = 600 and p = 1/6, giving mp = 600(1/6) = 100. Th is is aninteger; thus, 100 is the point prediction of Y . With the help of the website calculator (detailsnot given), I find that P (Y = 100) = 0.0437. For comp arison, P (Y = 99) = 0.0436 andP (Y = 101) = 0.0432. Thus, if 99 is your life-long favorite num ber, it is difficult for me tocriticize using it as you r point prediction. In the long-run, you will have one fewer correctpoint prediction for every 10,00 0 t imes you cast the blue die 600 t imes. That’s a lot of diecasting!• Suppose that m = 200 and p = 0.50. Then, mp = 200(0.5) = 100 is an integer; thus, 100 isthe point prediction of Y . With the help o f the websit e calculator, I find that P (Y = 100) =0.0563.• Suppose that m = 300 and p = 0.30. Then, mp = 300(0.3) = 90 is an integer; thus, 90 isthe point prediction of Y . With the help of the websit e calculator, I find that P (Y = 90) =0.0502.• Suppose that m = 20 and p = 0.42. Then, mp = 20(0.42) = 8.4 i s not an integer. Themost likely value of Y is either 8 or 9. With the help of the website calculator, I find thatP (Y = 8) = 0.1768 and P (Y = 9) = 0.1707. Thus, ˆy = 8.• Suppose that m = 75 and p = 0.50. Then, mp = 75(0.50) = 37.5 is not an integer. Themost likely value of Y is either 37 or 38. With the help of the website calculator, I find thatP (Y = 37) = 0.0912 and P (Y = 38) = 0.0912. Because these probabilities are identical, Ichoose the even integer; thus, ˆy = 38.• Suppose that m = 100 and p = 0.615. Then, mp = 100(0.615) = 61.5 i s not an integer.The most likely value of Y is either 6 1 or 62. With the help of the website calculator, I findthat P (Y = 61) = 0.0811 and P (Y = 62) = 0.0815. Thus, ˆy = 62.In each of the above examples we saw that the probability that a poin t prediction i s correctis very small. As a result, scientists usually prefer a prediction interval. It is possible to create aone-sided prediction interval, …
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