9F (Probability)

Independence gives

E E (Z) = \sum_{z, y} F (y, z) P (Y = y) P (Z = z) = E F (Y, Z)

. Moreover,

E V (Z) = E (F (Y, Z)^{2}) - E (E (Z)^{2})

, while

Var E (Z) = E (E (Z)^{2}) - (E F (Y, Z))^{2}

Var F (Y, Z) = E V (Z) + Var E (Z) .

b

For one Bernoulli variable, direct expansion gives

Var F (X_{1}) = p (1 - p) (F (1) - F (0))^{2}

. Applying the law of total variance successively to the coordinates gives the variance tensorization

Var F (X) \leq p (1 - p) i = 1 \sum n E (F (X) - F (X^{i}))^{2} .

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