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That s as far as we can reduce the matrix, but it s quite an improvement! This is not a coincidence, even though it may appear that way at first In the matrix morphing process, we did a lot of multiplying Those multiples went along for the ride, inflating the numbers They have served their purpose It s good to be rid of them 7 To reduce the matrix in solution to Prob 6 to unit diagonal form, we divide each row by 11 That gives us

When two random variables are independent and have the same marginal densities, they are independent and identically distributed (iid) Example 47 The random variables u and v in Example 46 are independent There is a result called the central limit theorem [31, 107] that states that if

1 0 0 0 1 0 0 0 1 89/11 20/11 58/11

while the third digit, called the multiplier, corresponds to the number of zeros to be added to the first two digits

(420)

The solution to the original linear system, stated at the end of solution to Prob 3, is apparently x = 89/11 y = 20/11 z = 58/11

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8 Here are the equations from solution to Prob 3, which we hope we have solved as a linear system: 4y z = 2 5x (3/2)y + 8z = 1 x+y+z=1 Let s check the first equation with the values from the solution to Prob 7: 4y z = 2 4 ( 20/11) ( 58/11) = 2 80/11 + 58/11 = 2 22/11 = 2 2 = 2 Checking in the second equation: 5x (3/2)y + 8z = 1 5 89/11 [(3/2) ( 20/11)] + 8 ( 58/11) = 1 445/11 ( 30/11) 464/11 = 1 445/11 + 30/11 464/11 = 1 11/11 = 1 1=1 Checking in the third equation: x+y+z=1 89/11 + ( 20/11) + ( 58/11) = 1 89/11 20/11 58/11 = 1 11/11 = 1 1=1 We can now be confident that the solution to the three-by-three linear system is x = 89/11 y = 20/11 z = 58/11 9 Here s the three-by-three linear system we have been told to describe as a matrix: x+y+z=1 x+y+z=2 x+y+z=3

where the xi are independent random variables, then as N increases the distribution for xN approaches a Gaussian distribution, independent of the distributions for the individual xi This rather remarkable result motivates the the importance of Gaussian random variables in applications Whenever a random e ect is the superposition of many small random a ects, the superimposed e ect can be accurately modeled as a Gaussian random variable When two random variables are not independent, it is useful to have metrics to quantify the amount of interdependence Two important metrics are correlation and covariance The correlation matrix between two random variables v and w is de ned by cor(v, w) = E vw = VW p(V, W) dV dW (421)

1st Significant Figure 2nd Significant Figure Multiplier Tolerance 01 F 10% CSGNetworkCom 6/4/92

19

The covariance matrix for two vector valued random variables v and w is de ned by cov(v, w) = = E (v v )(w w ) (V v )(W w ) p(V, W) dV dW (422)

1 1 1 1 1 1 1 1 1 1 2 3

The correlation coe cient vw is a normalized measure of the correlation between the two scalar random variables v and w that is de ned as vw = cov(v, w) v w (423)

104 k

Now let s try to get this into unit diagonal form The first step along the way is to seek the echelon form We can start by doing one of three things: Make the first entry in the second row vanish Make the first entry in the third row vanish Make the second entry in the third row vanish This is easy too easy! Suppose we want to make the first entry in the third row vanish We can multiply the first row through by 1, getting

1 1 1 1 1 1 1 1 1 1 2 3

The correlation coe cient always satis es 1 vw 1 When the magnitude of vw is near one, then knowledge of one of the random variables will

1 1 0 1 1 0 1 1 0 1 2 2

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