Locating the Bernstein crossover

NumericalIntermediate⏱ 30 minHigh-Dim Probabilitybernsteinhoeffdingcrossovercomparison

Problem

Let $X_1,\dots,X_n$ be i.i.d. Bernoulli $(p)$ with $p = 0.01$ , centred, so $|X_i|\le1$ and $\sigma^2 = np(1-p)$ .

At which deviation $t$ do the two terms $2\sigma^2$ and $\tfrac23 Mt$ in Bernstein's denominator become equal?
For $n = 10^4$ and $t = 20$ , compare the Hoeffding and Bernstein bounds numerically.
Explain in one sentence when each inequality is the right tool.

Compute the requested quantities and say what they mean. A number without an interpretation is not an answer.

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