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Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 2 / 12F / i / Solution

The Chebyshev inequality states that for a random variable

Y

with finite variance and every

δ > 0

,

P (∣ Y - E Y ∣ \geq δ) \leq \frac{var Y}{δ ^{2}} .

Indeed, Markov inequality applied to the nonnegative variable

(Y - E Y)^{2}

gives

P ((Y - E Y)^{2} \geq δ^{2}) \leq \frac{E [( Y - E Y ) ^{2} ]}{δ ^{2}} .

The sum

S_{n} = X_{1} + \dots + X_{n}

has the binomial distribution, so

B_{n} (p) = k = 0 \sum n f (k / n) (k n) p^{k} (1 - p)^{n - k} .

For fixed

n

this is a finite sum of polynomial functions of

p

, hence is a polynomial.

Solved by gpt-5.6-sol high.

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