Nous proposons ici de démontrer la formule de calcul de la covariance. On a par définition :
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1 |
n |
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cov(x,y) |
= |
––– |
S |
( xi
– mx)( yi – my ) |
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n |
i = 1 |
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On développe les produits :
( xi
– mx)( yi – my ) |
= |
xi yi – xi my
+- mx yi + mx my |
D’où la somme :
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1 |
n |
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1 |
n |
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1 |
n |
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1 |
n |
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cov(x, y) |
= |
––– |
S |
xi yi – |
––– |
S |
xi my – |
––– |
S |
mx yi + |
––– |
S |
mx my |
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n |
i = 1 |
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n |
i = 1 |
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n |
i = 1 |
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n |
i = 1 |
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On met les termes constants en facteurs. On a alors :
1 |
n |
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1 |
n |
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|||||||
––– |
S |
xi my |
= |
my |
––– |
S xi |
= |
mx my |
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n |
i = 1 |
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n |
i = 1 |
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|||||||
1 |
n |
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1 |
n |
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––– |
S |
mx yi |
= |
mx |
––– |
S yi |
= |
my mx |
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n |
i = 1 |
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n |
i = 1 |
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1 |
n |
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1 |
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––– |
S |
mx my |
= |
mx my |
––– |
x n |
= |
mx my |
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n |
i = 1 |
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n |
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On obtient :
1 |
n |
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1 |
n |
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––– |
S |
( xi – mx)( yi
– my ) |
= |
––– |
S |
xi yi – mx my
– mx my + mx my |
n |
i = 1 |
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|
n |
i = 1 |
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D’où finalement :
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1 |
n |
|
cov(x,y) |
= |
––– |
S |
xi yi – mx my |
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|
n |
i = 1 |
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