Data Science in Theory and Practice. Maria Cristina Mariani

Data Science in Theory and Practice

Row 1st Column Blank 2nd Column Variable 1 left-parenthesis dollar sales right-parenthesis colon 3rd Column 48 22 50 2nd Row 1st Column Blank 2nd Column Variable 2 left-parenthesis number of movies right-parenthesis colon 3rd Column 3 1 2 EndLayout"/>

Then the data matrix bold upper X is

bold upper X equals Start 3 By 2 Matrix 1st Row 1st Column 48 2nd Column 3 2nd Row 1st Column 22 2nd Column 1 3rd Row 1st Column 50 2nd Column 2 EndMatrix comma

with three rows and two columns.

We now present some descriptive statistics. We will begin with the mean vectors.

3.3 Mean Vectors

Let bold upper X Superscript upper T Baseline equals left-bracket x 1 comma x 2 comma ellipsis comma x Subscript p Baseline right-bracket be random variables, where upper T denotes the transpose of a matrix. The expected value of the random p times 1 vector is defined as the vector of expectations i.e.

upper E left-parenthesis bold upper X right-parenthesis equals upper E Start 4 By 1 Matrix 1st Row x 1 2nd Row x 2 3rd Row vertical-ellipsis 4th Row x Subscript p Baseline EndMatrix equals Start 4 By 1 Matrix 1st Row upper E left-parenthesis x 1 right-parenthesis 2nd Row upper E left-parenthesis x 2 right-parenthesis 3rd Row vertical-ellipsis 4th Row upper E left-parenthesis x Subscript p Baseline right-parenthesis EndMatrix period

More generally, if bold upper Z Subscript n times p Baseline equals left-bracket z Subscript j k Baseline right-bracket is a matrix of random variables, then the upper E left-parenthesis bold upper Z right-parenthesis is the matrix of expectations with elements left-bracket upper E left-parenthesis z Subscript j k Baseline right-parenthesis right-bracket , i.e.:

StartLayout 1st Row 1st Column bold upper Z 2nd Column equals upper E Start 6 By 6 Matrix 1st Row 1st Column z Subscript 1 comma 1 Baseline 2nd Column z Subscript 1 comma 2 Baseline 3rd Column midline-horizontal-ellipsis 4th Column z Subscript 1 comma k Baseline 5th Column midline-horizontal-ellipsis 6th Column z Subscript 1 comma p Baseline 2nd Row 1st Column z Subscript 2 comma 1 Baseline 2nd Column z Subscript 2 comma 2 Baseline 3rd Column midline-horizontal-ellipsis 4th Column z Subscript 2 comma k Baseline 5th Column midline-horizontal-ellipsis 6th Column z Subscript 2 comma p Baseline 3rd Row 1st Column vertical-ellipsis 2nd Column vertical-ellipsis 3rd Column Blank 4th Column vertical-ellipsis 5th Column vertical-ellipsis 6th Column vertical-ellipsis 4th Row 1st Column z Subscript j comma 1 Baseline 2nd Column z Subscript j comma 2 Baseline 3rd Column midline-horizontal-ellipsis 4th Column z Subscript j comma k Baseline 5th Column midline-horizontal-ellipsis 6th Column z Subscript j comma p Baseline 5th Row 1st Column vertical-ellipsis 2nd Column vertical-ellipsis 3rd Column Blank 4th Column vertical-ellipsis 5th Column vertical-ellipsis 6th Column vertical-ellipsis 6th Row 1st Column z Subscript n comma 1 Baseline 2nd Column z Subscript n comma 2 Baseline 3rd Column midline-horizontal-ellipsis 4th Column z Subscript n comma k Baseline 5th Column midline-horizontal-ellipsis 6th Column z Subscript n comma p Baseline EndMatrix 2nd Row 1st Column Blank 2nd Column equals Start 6 By 6 Matrix 1st Row 1st Column upper E left-parenthesis z Subscript 1 comma 1 Baseline right-parenthesis 2nd Column upper E left-parenthesis z Subscript 1 comma 2 Baseline right-parenthesis 3rd Column midline-horizontal-ellipsis 4th Column upper E left-parenthesis z Subscript 1 comma k Baseline right-parenthesis 5th Column midline-horizontal-ellipsis 6th Column upper E left-parenthesis z Subscript 1 comma p Baseline right-parenthesis 2nd Row 1st Column upper E left-parenthesis z Subscript 2 comma 1 Baseline right-parenthesis 2nd Column upper E left-parenthesis z Subscript 2 comma 2 Baseline right-parenthesis 3rd Column midline-horizontal-ellipsis 4th Column upper E left-parenthesis z Subscript 2 comma k Baseline right-parenthesis 5th Column midline-horizontal-ellipsis 6th Column upper E left-parenthesis z Subscript 2 comma p Baseline right-parenthesis 3rd Row 1st Column vertical-ellipsis 2nd Column vertical-ellipsis 3rd Column Blank 4th Column vertical-ellipsis 5th Column vertical-ellipsis 6th Column vertical-ellipsis 4th Row 1st Column upper E left-parenthesis z Subscript j comma 1 Baseline right-parenthesis 2nd Column upper E left-parenthesis z Subscript j comma 2 Baseline right-parenthesis 3rd Column midline-horizontal-ellipsis 4th Column upper E left-parenthesis z Subscript j comma k Baseline right-parenthesis 5th Column midline-horizontal-ellipsis 6th Column upper E left-parenthesis z Subscript j comma p Baseline right-parenthesis 5th Row 1st Column vertical-ellipsis 2nd Column vertical-ellipsis 3rd Column Blank 4th Column vertical-ellipsis 5th Column vertical-ellipsis 6th Column vertical-ellipsis 6th Row 1st Column upper E left-parenthesis z Subscript n comma 1 Baseline right-parenthesis 2nd Column upper E left-parenthesis z Subscript n comma 2 Baseline right-parenthesis 3rd Column midline-horizontal-ellipsis 4th Column upper E left-parenthesis z Subscript n comma k Baseline right-parenthesis 5th Column midline-horizontal-ellipsis 6th Column upper E left-parenthesis z Subscript n comma p Baseline right-parenthesis EndMatrix period EndLayout

For a random vector bold upper X Superscript upper T Baseline equals left-bracket x 1 comma x 2 comma ellipsis comma x Subscript p Baseline right-bracket , the mean vector consists of the means of each variable:

where upper E left-parenthesis upper X Subscript i Baseline right-parenthesis equals mu Subscript i . For example let x 11 comma x 21 comma ellipsis comma x Subscript n Baseline 1 Baseline be measurements on the first variable. Then the sample mean for the first variable x overbar Subscript 1 is defined as follows:

x overbar Subscript 1 Baseline equals StartFraction 1 Over n EndFraction sigma-summation Underscript j equals 1 Overscript n Endscripts x Subscript j Baseline 1 Baseline period

The sample mean can be computed from the measurements on each of the variables. Therefore, in general for sample means, we have:

x overbar Subscript k Baseline equals StartFraction 1 Over n EndFraction sigma-summation Underscript j equals 1 Overscript n Endscripts x Subscript j k Baseline comma k equals 1 comma 2 comma ellipsis <hr><noindex><a href=

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