The Distribution of a Linear Combination of Random Variables Bernd - PowerPoint PPT Presentation

The Distribution of a Linear Combination of Random Variables Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Introduction logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Introduction This section mostly summarizes earlier results in a slightly more abstract light. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Definition. Let X 1 ,..., X n be random variables and let a 1 ,..., a n be numbers. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Definition. Let X 1 ,..., X n be random variables and let a 1 ,..., a n be numbers. Then the random variable n ∑ Y = a k X k k = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Definition. Let X 1 ,..., X n be random variables and let a 1 ,..., a n be numbers. Then the random variable n ∑ Y = a k X k = a 1 X 1 + ··· + a n X n k = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Definition. Let X 1 ,..., X n be random variables and let a 1 ,..., a n be numbers. Then the random variable n ∑ Y = a k X k = a 1 X 1 + ··· + a n X n k = 1 is called a linear combination of X 1 ,..., X n . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Definition. Let X 1 ,..., X n be random variables and let a 1 ,..., a n be numbers. Then the random variable n ∑ Y = a k X k = a 1 X 1 + ··· + a n X n k = 1 is called a linear combination of X 1 ,..., X n . Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Definition. Let X 1 ,..., X n be random variables and let a 1 ,..., a n be numbers. Then the random variable n ∑ Y = a k X k = a 1 X 1 + ··· + a n X n k = 1 is called a linear combination of X 1 ,..., X n . n 1 ∑ Example. X = nX k is a linear combination of X 1 ,..., X k . k = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Proposition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Proposition. Let X 1 ,..., X n be random variables with means µ 1 ,..., µ n and standard deviations σ 1 ,..., σ n . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Proposition. Let X 1 ,..., X n be random variables with means µ 1 ,..., µ n and standard deviations σ 1 ,..., σ n . � � n ∑ 1. E a k X k k = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Proposition. Let X 1 ,..., X n be random variables with means µ 1 ,..., µ n and standard deviations σ 1 ,..., σ n . � � n n ∑ ∑ = a k E ( X k ) 1. E a k X k k = 1 k = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Proposition. Let X 1 ,..., X n be random variables with means µ 1 ,..., µ n and standard deviations σ 1 ,..., σ n . � � n n n ∑ ∑ ∑ = a k E ( X k ) = 1. E a k X k a k µ k . k = 1 k = 1 k = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Proposition. Let X 1 ,..., X n be random variables with means µ 1 ,..., µ n and standard deviations σ 1 ,..., σ n . � � n n n ∑ ∑ ∑ = a k E ( X k ) = 1. E a k X k a k µ k . k = 1 k = 1 k = 1 � � n ∑ 2. V a k X k k = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Proposition. Let X 1 ,..., X n be random variables with means µ 1 ,..., µ n and standard deviations σ 1 ,..., σ n . � � n n n ∑ ∑ ∑ = a k E ( X k ) = 1. E a k X k a k µ k . k = 1 k = 1 k = 1 � � n n n ∑ ∑ ∑ 2. V a k X k = a j a k Cov ( X j , X k ) . k = 1 j = 1 k = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Proposition. Let X 1 ,..., X n be random variables with means µ 1 ,..., µ n and standard deviations σ 1 ,..., σ n . � � n n n ∑ ∑ ∑ = a k E ( X k ) = 1. E a k X k a k µ k . k = 1 k = 1 k = 1 � � n n n ∑ ∑ ∑ 2. V a k X k = a j a k Cov ( X j , X k ) . k = 1 j = 1 k = 1 3. If X 1 ,..., X n are independent logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Proposition. Let X 1 ,..., X n be random variables with means µ 1 ,..., µ n and standard deviations σ 1 ,..., σ n . � � n n n ∑ ∑ ∑ = a k E ( X k ) = 1. E a k X k a k µ k . k = 1 k = 1 k = 1 � � n n n ∑ ∑ ∑ 2. V a k X k = a j a k Cov ( X j , X k ) . k = 1 j = 1 k = 1 3. If X 1 ,..., X n are independent, then � � n ∑ V a k X k k = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Proposition. Let X 1 ,..., X n be random variables with means µ 1 ,..., µ n and standard deviations σ 1 ,..., σ n . � � n n n ∑ ∑ ∑ = a k E ( X k ) = 1. E a k X k a k µ k . k = 1 k = 1 k = 1 � � n n n ∑ ∑ ∑ 2. V a k X k = a j a k Cov ( X j , X k ) . k = 1 j = 1 k = 1 3. If X 1 ,..., X n are independent, then � � n n a 2 ∑ ∑ = k V ( X k ) V a k X k k = 1 k = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Proposition. Let X 1 ,..., X n be random variables with means µ 1 ,..., µ n and standard deviations σ 1 ,..., σ n . � � n n n ∑ ∑ ∑ = a k E ( X k ) = 1. E a k X k a k µ k . k = 1 k = 1 k = 1 � � n n n ∑ ∑ ∑ 2. V a k X k = a j a k Cov ( X j , X k ) . k = 1 j = 1 k = 1 3. If X 1 ,..., X n are independent, then � � n n n a 2 a 2 k σ 2 ∑ ∑ ∑ = k V ( X k ) = V a k X k k . k = 1 k = 1 k = 1 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Proof of Part 1. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Proof of Part 1. We know that E ( X + Y ) = E ( X )+ E ( Y ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Proof of Part 1. We know that E ( X + Y ) = E ( X )+ E ( Y ) and it is easy to see that E ( aX ) = aE ( X ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Proof of Part 1. We know that E ( X + Y ) = E ( X )+ E ( Y ) and it is easy to see that E ( aX ) = aE ( X ) . (Remember that expected values are sums or integrals logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Proof of Part 1. We know that E ( X + Y ) = E ( X )+ E ( Y ) and it is easy to see that E ( aX ) = aE ( X ) . (Remember that expected values are sums or integrals and constants can be factored out of sums and integrals.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables

Proof of Part 1. We know that E ( X + Y ) = E ( X )+ E ( Y ) and it is easy to see that E ( aX ) = aE ( X ) . (Remember that expected values are sums or integrals and constants can be factored out of sums and integrals.) E ( a 1 X 1 + ··· a n − 1 X n − 1 + a n X n ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Distribution of a Linear Combination of Random Variables