371chapter7f2011

# 371chapter7f2011 - Chapter 7 Rules for Means and Variances...

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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 .A n dd r y .B u ti ti su s e f u lf o ru n d e r - standing many of the methods we will learn later in this course. We have random variables X 1 ,X 2 ,...X n .Th roughou tth issec t ion ,wew i l lassumetha tthese random variables are independent. Sometimes they will also be identically distributed, but we don’t need identically distributed for our main result. (There isasimilarresultwithoutindependencetoo, but we won’t need it.) Let μ i denote the mean of X i .Le t σ 2 i denote the variance of X i . Let b 1 ,b 2 ,...,b n denote n numbers. DeFne 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 .I fthesequencei si . i .d .thenwecanw r i te μ = μ i and σ 2 = σ 2 i .Inth i sca sethe 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 ,thenweusuallycalltherandomvariables X and Y instead of X 1 and X 2 .W ege t 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

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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 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 intervalpredictions. 7.2.1 When p is Known We begin with point prediction of Y ;wedeno tethepo in tpred ic t ionby ˆ y .W eadop tthec r i te r ion that we want the probability of being correct to be as large as possible. Below is the result. Calculate the mean of Y ,whichis mp .If mp is an integer then it is the most probable value of Y and our prediction is ˆ y = mp .Herearesomeexamp les . Suppose that m =20 and p =0 . 50 .Then , mp =20(0
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371chapter7f2011 - Chapter 7 Rules for Means and Variances...

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