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IEN310 Chapter 6

# IEN310 Chapter 6 - Chapter 6 Sums of Random Variables 6...

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Chapter 6: Sums of Random Variables 6. Introduction There are many applications of probability theory in which random variables of the form W n = X 1 + · · · X n appear. Our goal is to derive the n -dimensional probability model of W n . First, we will consider expected values related to W n rather than a complete model of W n . There are many applications where this information is essentially all we need. We will then consider techniques that allow us to derive a complete model of W n when X 1 , . . . , X n are mutually independent. As we shall see, a useful way to analyze the sum of independent random variables is to transform the PDF or PMF of each random variable X 1 , . . . , X n to a moment generating function . 6.1 Expected Values of Sums Section 4.7 in Chapter 4 addressed computing the expected values and variances of pairs of random variables. Those theorems can be generalized to describe the expected values and variances of sums of random variables. Generalizing Theorem 4.14 results in the following theorem, which essentially states that the expected value of the sum equals the sum of the expected values, whether or not X 1 , . . . , X n are independent. Theorem 6.1 For any set of random variables X 1 , . . . , X n , the expected value of W n = X 1 + · · · + X n is E [ W n ] = E [ X 1 ] + E [ X 2 ] + · · · + E [ X n ] . Similarly, generalizing Theorem 4.15 results in the following theorem regarding the vari- ance of a sum of random variables. Theorem 6.2 The variance of W n = X 1 + · · · + X n is Var[ W n ] = n X i =1 Var[ X i ] + 2 n - 1 X i =1 n X j = i +1 Cov[ X i , X j ] . Note that in terms of the random vector X = [ X 1 · · · X n ] 0 , Var[ W n ] is the sum of all the elements in the covariance matrix C X . The diagonal of this matrix consists of all the Var[ X i ] = Cov[ X i , X j ] terms, and the off-diagonal elements of this matrix are the Cov[ X i , X j ] = Cov[ X j , X i ] terms. The variance of the sum of of the random variables X 1 , . . . , X n becomes simplified when X 1 , . . . , X n are uncorrelated, as stated in the following theorem, because under these conditions Cov[ X i , X j ] = 0 for i 6 = j . Theorem 6.3 When X 1 , . . . , X n are uncorrelated , Var[ W n ] = Var[ X 1 ] + · · · + Var[ X n ] . 89

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In the following examples, we make use of a technique whereby we denote the random variable X i as an indicator variable such that X i = ( 1 some particular event i occurs , 0 otherwise. Example 1 (pages 245-246) At a party of n 2 people, each person throws a hat in a common box. The box is shaken and each person blindly draws a hat from the box without replacement . A match occurs if a person draws his or her own hat. Let V n denote the number of matches. (a) Determine E [ V n ] and Var[ V n ], the expected value and variance of the number of matches. (b) Suppose after drawing a hat, each person immediately returns to the box the hat that he or she drew, prior to the next person’s draw of a hat—that is, the hats are now being drawn with replacement . Determine the expected value and variance of the number of matches.
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