Sample Mean and Sample Covariance - Weighted Samples

Weighted Samples

It has been suggested that portions of this section be moved into Weighted mean. (Discuss)

In a weighted sample, each vector (each set of single observations on each of the K random variables) is assigned a weight . Without loss of generality, assume that the weights are normalized:

(If they are not, divide the weights by their sum). Then the weighted mean vector is given by

and the elements of the weighted covariance matrix are

q_{jk}=\frac{\sum_{i=1}^{N}w_i}{\left(\sum_{i=1}^{N}w_i\right)^2-\sum_{i=1}^{N}w_i^2} \sum_{i=1}^N w_i \left( x_{ij}-\bar{x}_j \right) \left( x_{ik}-\bar{x}_k \right) .

If all weights are the same, the weighted mean and covariance reduce to the sample mean and covariance above.

Read more about this topic:  Sample Mean And Sample Covariance

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