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確率密度関数と期待値と分散について

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前提

確率密度関数を\(f(x)\)とする

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期待値

離散確率

\[E[X] = \sum x f(x)\]

連続確率

\[E[X] = \int x f(x) dx\]

期待値の性質

\[E[X + c] = E[X] + c\] \[E[a X] = a E[X]\] \[E[X + Y] = E[X] + E[Y]\]

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分散

連続確率

\[V[X] = \int (x-\mu)^2f(x)dx\]

分散の定義の一般形

\[V[X] = E[X^2] - {E[X]}^2\]

分散の性質

\[V[X + c] = V[X]\] \[V[a X] = a^2 V[X]\] \[V[X + Y] = V[X] + V[Y] + 2Cov[X, Y]\]

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参考

• 分散と共分散/TauStation