共分散の基本的性質

by nomura · 2020年8月18日

X, Y

を確率変数、

a, b

を定数とする。

(1)

C o v (X, Y) = C o v (Y, X)

C o v (X, Y + a) = C o v (X, Y)

C o v (X, a Y) = a C o v (X, Y)

\begin{aligned} C o v (X, Y) & = E ((X - E (X)) (Y - E (Y))) \\ = C o v (Y, X) \end{aligned}

\begin{aligned} C o v (X, Y + a) & = E (X (Y + a)) - E (X) E (Y + a) \\ = E (X Y) + a E (X) - E (X) E (Y) + a E (X) \\ = E (X Y) - E (X) E (Y) \\ = C o v (X, Y) \end{aligned}

\begin{aligned} C o v (X, a Y) & = E (X a Y) - E (X) E (a Y) \\ = a (E (X Y) - E (X) E (Y)) \\ = a C o v (X, Y) \end{aligned}

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タイトル	共分散の基本的性質
URL	https://www.nomuramath.com/ugczz9jg/
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