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Unformatted text preview: Chapter 7 Rules for Means and Variances; Prediction 7.1 Rules for Means and Variances The material in this section is very technical and algebraic. And dry. But it is useful for under- standing many of the methods we will learn later in this course. We have random variables X 1 , X 2 , . . .X n . Throughout this section, we will assume that these rv’s are independent. Sometimes they will also be identically distributed, but we don’t need i.d. for our main result. (There is a similar result without independence too, but we won’t need it.) Let μ i denote the mean of X i . Let σ 2 i denote the variance of X i . Let b 1 , b 2 , . . ., b n denote n numbers. Define W = b 1 X 1 + b 2 X 2 + . . .b n X n . W is a linear combination of the X i ’s. The main result is • The mean of W is μ W = ∑ n i =1 b i μ i . • The variance of W is σ 2 W = ∑ n i =1 b 2 i σ 2 i . Special Cases 1. i.i.d. case . If the sequence is i.i.d. then we can write μ = μ i and σ 2 = σ 2 i . In this case the mean of W is μ W = ( ∑ n i =1 b i ) μ and the variance of W is σ 2 W = ( ∑ n i =1 b 2 i ) σ 2 . 2. Two independent rv’s. If n = 2 , then we usually call them X and Y instead of X 1 and X 2 . We get W = b 1 X + b 2 Y which has mean μ W = b 1 μ X + b 2 μ Y and variance σ 2 W = b 2 1 σ 2 X + b 2 2 σ 2 Y . 3. Two i.i.d. rv’s. Combining the notation of the previous two items, W = b 1 X + b 2 Y has mean μ W = ( b 1 + b 2 ) μ and variance σ 2 W = ( b 2 1 + b 2 2 ) σ 2 . Especially important is the case W = X + Y which has mean μ W = 2 μ and variance σ 2 W = 2 σ 2 . Another important case is W = X- Y which has mean μ W = 0 and variance σ 2 W = 2 σ 2 . 71 7.2 Predicting for Bernoulli Trials Predictions are tough, especially about the future—Yogi Berra. We plan to observe m BT and want to predict the total number of successes that we will get. Let Y denote the r.v. 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. 7.2.1 When p is Known We begin with point prediction of Y . We adopt the criterion that we want the probability of being correct to be as large as possible. Below is the result. Calculate the mean of Y , which is mp . If mp is an integer then it is the most probable value of Y and our prediction is ˆ y = mp . Here are some examples. • Suppose that m = 20 and p = 0 . 50 . Then, mp = 20(0 . 5) = 10 is an integer, so 10 is our point prediction of Y . With the help of our website calculator (details not given), we find that P ( Y = 10) = 0 . 1762 . • Suppose that m = 200 and p = 0 . 50 . Then, mp = 200(0 . 5) = 100 is an integer, so 100 is our point prediction of Y . With the help of our website calculator, we find that P ( Y = 100) = 0 . 0563 ....
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