Independent daily observations each lie in an interval of width w = 7. Hoeffding's inequality gives P(|X̄ − μ| ≥ ε) ≤ 2 exp(−2nε²/w²). With ε = 1, what is the smallest whole n for which this bound is at most 2 exp(−3)? Give a whole number.

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Days for a Hoeffding guarantee

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Independent daily observations each lie in an interval of width $w = 7$ . Hoeffding's inequality gives $P (∣ X ˉ - μ ∣\geq ϵ) \leq 2 exp (- \frac{2 n ϵ ^{2}}{w ^{2}})$ . With $ϵ = 1$ , what is the smallest whole $n$ for which this bound is at most $2 exp (- 3)$ ? Give a whole number.

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Hint

Compare the exponents: $\frac{2 n ϵ ^{2}}{w ^{2}}$ must be at least $3$ .

Solution

We need $\frac{2 n}{49} \geq 3$ , so $n \geq 3 \times \frac{49}{2}$ , and $3 \times \frac{49}{2} = 73.5$ .
Rounding up gives $n = 74$ .

Answer: $74$

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