Chapter 14 Rules for Means and Variances Prediction 14 1 Rules for Means and Variances The result in this section is very technical and algebraic And dry But it is useful for understanding many of prediction results we obtain in this course beginning later in this chapter We have independent random variables W1 and W2 Note that typically they are not identically distributed The result below is an extremely special case of a much more general result It will suffice however for our needs thus I see no reason to subject you to the pain of viewing the general result We need some notation Let 1 2 denote the mean of W1 W2 Let Var W1 Var W2 denote the variance of W1 W2 Let b denote any number Define W W1 bW2 Result 14 1 The mean and variance of W For the notation given above The mean of W is The variance of W is W 1 b 2 14 1 Var W Var W1 b2 Var W2 14 2 In our two applications of this result in this chapter the number b will be taken to equal 1 2 thus W 1 1 2 2 1 1 0 for our applications 337 14 2 Predicting for Bernoulli Trials Predictions are tough especially about the future Yogi Berra We plan to observe m Bernoulli trials and want to predict the total number of successes that will be obtained Let Y denote the random variable and y the observed value of the total number of successes in the future m trials Similar to estimation we will learn about point and interval predictions of the value of Y 14 2 1 When p is Known Because prediction is a new idea in this course I want to present a gentle introduction to it Suppose that you have a favorite pair of dice one colored blue and the other white Let s focus on the blue die And let s say your favorite number is 6 perhaps you play Risk a great deal in Risk 6 is the best 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 will obtain You believe that the die is balanced i e that the six possible outcomes are equally likely to occur OK Quick Don t think of any of the wonderful things you have learned in this course Give me your point single number prediction of how many 6 s you will obtain I conjecture that your answer is 100 I asked this question several times over the years to a live lecture and always save once received the answer 100 from the student who volunteered to answer One year a guy said 72 and got a big laugh I failed him because it is my job to make the jokes such as they are No I didn t really fail him but I was more than a bit annoyed that he got a larger laugh than I did with my much cleverer anecdotes My academic grandfather my advisor s advisor who happened to be male is Herb Robbins a very brilliant and witty man Herb was once asked what mathematical statisticians do and he replied They find out what non statisticians do and prove it s optimal Thus I am going to argue that your answer of 100 is the best answer to the die question I posed above In order to show that something is best mathematically we find a way to measure good and whichever candidate answer has the largest amount of good is best This is the approach for the glass half full people More often one finds a way to measure bad and whichever candidate answer 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 by the single number y We adopt the criterion that we want the probability of being correct to be as large as possible Thus we define being correct as good and seek to maximize the probability that we will get a good result Result 14 2 The best point prediction of Y Calculate the mean of Y which is mp If mp is an integer then it is uniquely the most probable value of Y and our point prediction is y mp 338 If mp is not an integer then the most probable value of Y is one of the integers immediately on either side of mp Check them both whichever is more probable 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 This is an integer thus 100 is the point prediction of Y With the help of the website calculator details not given I find that P Y 100 0 0437 For comparison P Y 99 0 0436 and P Y 101 0 0432 Thus if 99 is your life long favorite number it is difficult for me to criticize using it as your point prediction In the long run you will have one fewer correct point prediction for every 10 000 times you cast the blue die 600 times That s a lot of die casting Suppose that m 200 and p 0 50 Then mp 200 0 5 100 is an integer thus 100 is the point prediction of Y With the help of the website 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 is the point prediction of Y With the help of the website calculator I find that P Y 90 0 0502 Suppose that m 20 and p 0 42 Then mp 20 0 42 8 4 is not an integer The most likely value of Y is either 8 or 9 With the help of the website calculator I find that P Y 8 0 1768 and P Y 9 0 1707 Thus y 8 Suppose that m 75 and p 0 50 Then mp 75 0 50 37 5 is not an integer The most likely value of Y is either 37 or 38 With the help of the website calculator I find that P Y 37 0 0912 and P Y 38 0 0912 Because these probabilities are identical I choose the even integer thus y 38 Suppose that m 100 and p 0 615 Then mp 100 0 615 61 5 is not an integer The most likely value of Y is either 61 or 62 With the help of the website calculator I find that P Y 61 0 0811 and P Y 62 0 0815 Thus y 62 In each of the above examples we saw that the probability that a point prediction is correct is very small As a result scientists usually prefer a prediction …
View Full Document
Unlocking...