Linear Algebra and Its Applications · Problem 3.3.30

Orthogonality — Gilbert Strang

Mathematics · statistics-variance-reduction

Problem

A doctor takes four readings of your heart rate. The best solution to

x = b_1, \ldots, x = b_4

is the average

\hat{x}

b_1, \ldots, b_4

. The matrix

A

is a column of 1s. Problem 29 gives the expected error

(\hat{x} - x)^2

\sigma^2 (A^T A)^{-1} =

\_\_\_\_. By averaging, the variance drops from

\sigma^2

\sigma^2/4