Data Science in Theory and Practice. Maria Cristina Mariani
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(3.12)
3.6.2 Linear Combinations of Sample Variance and Covariance
The sample variance of
(3.13)
We recall from Section 2.3 that variance is always nonnegative. Thus, we have
If we define another linear combination
(3.14)
where
Please refer to Johnson and Wichern (2014) for the proof of (3.14).
3.6.3 Linear Combinations of Sample Correlation
The sample correlation between
(3.15)
We note that the sample results in Section 3.6 have population counterparts. We briefly state them below:
The population mean of
where
where