Past exam of the options I course of the University of Oxford / 2009 / M / M1 / c

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- Solution c

Solution

The preceding marginal calculation makes each

V_{i}

uniformly distributed on

(0, 1)

, with expectation

1/2

1/12

. By independence,

E (S_{n} / n) = \frac{1}{2}, Var (S_{n} / n) = \frac{1}{12 n} .

Apply part (a) to the centred random variable

Z = S_{n} / n - 1/2

to obtain

P (∣ S_{n} / n - 1/2 ∣ \geq a) \leq \frac{1}{12 n a ^{2}} .

This is Chebyshev's inequality in this setting.

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