371chapter7f2011

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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 8

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online