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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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A finite Bernoulli model at n=100 and p=1/4

Physical scope

Exact specialization of the finite IID Bernoulli product model at n=100 and p=1/4; it is not a measured data set.

Example

In the explicitly stipulated product model of Finite Bernoulli fluctuations and matching exponential rates, take $n = 100$ and $p = 1 / 4$ . We calculate model probabilities and bounds without supplying any observed trial results.

Facts & Assumptions

Given: The finite IID Bernoulli model with $n = 100$ , $p = 1 / 4$ , and $K$ the number of successes.

[F1]

Finite Bernoulli fluctuations and matching exponential rates: under the explicit product law, $E K = n p$ , $Var (K) = n p (1 - p)$ , the two tail inequalities hold, and the point-mass bound has exponent $D (a ∥ p)$ .

Verification

technique · direct

Given: The model and hypotheses in the Example and [F1].

1.1givenF1algebra

Substitution gives $E K = 25$ and $Var (K) = 100 (1 / 4) (3 / 4) = 75 / 4$ . For $ε = 1 / 10$ , the Chebyshev bound is $(3 / 16) / (100 / 100) = 3 / 16$ , while the exponential bound is $2 e^{- 2}$ . These are upper bounds on a specified model probability, not results from 100 recorded trials.

2.1givenF1algebra∎

For $k = 35$ , $a = k / n = 7 / 20$ and $D (7 / 20 ∥ 1 / 4) = \frac{7}{20} \log \frac{7}{5} + \frac{13}{20} \log \frac{13}{15} .$ The point-mass estimate in [F1] therefore reads $\frac{1}{101} e^{- 100 D (7 / 20 ∥ 1 / 4)} \leq P_{1 / 4} (K = 35) \leq e^{- 100 D (7 / 20 ∥ 1 / 4)} .$ This finite example displays the polynomial prefactor and common exponential rate without numerical fitting or empirical inference.

Depends on

Finite Bernoulli fluctuations and matching exponential rates

A finite Bernoulli model at n=100 and p=1/4

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources