Binomial distribution ( $Bin (n, p)$ )

The binomial distribution counts successes in independent Bernoulli trials with common success probability.

Table of contents

Fourth centered moment of a binomial distribution ( $E (W - n p)^{4} = 3 n^{2} q^{2} + n q (1 - 6 q), q = p (1 - p)$ )

For

W \sim Bin (n, p)

, expand the fourth power of the sum of centered independent Bernoulli variables. Terms with an index occurring once vanish; single-index fourth moments contribute

n q (1 - 3 q)

and paired second moments contribute

3 n (n - 1) q^{2}

. Their sum is the displayed formula. Scaling by

n^{4}

yields a uniform

O (n^{- 2})

fourth moment for

W / n - p

Sample proportion ( $p = D / n$ )

 0  0

For independent Bernoulli observations with success probability

p

, the sample proportion is the success count divided by the sample size. It is an unbiased estimator with variance

p (1 - p) / n

. It supplies the arm-specific risks in a risk ratio and the mean difference in a two-proportion comparison.

Binomial likelihood

 0  0

x

successes occur in

n

independent Bernoulli trials, the binomial likelihood for the success probability

p

is proportional to

p^{x} (1 - p)^{n - x}

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The Binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent Bernoulli trials (experiments with two possible outcomes, often termed "success" and "failure"). This type of distribution is particularly useful in situations where you want to determine the likelihood of a certain number of successes within a series of trials.

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Binomial distribution ( $Bin (n, p)$ )

Fourth centered moment of a binomial distribution ( $E (W - n p)^{4} = 3 n^{2} q^{2} + n q (1 - 6 q), q = p (1 - p)$ )

Sample proportion ( $p = D / n$ )

Binomial likelihood

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Binomial distribution (Bin(n,p))

Fourth centered moment of a binomial distribution (E(W−np)4=3n2q2+nq(1−6q),q=p(1−p))

Sample proportion (p​=D/n)

Binomial likelihood

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Binomial distribution ( $Bin (n, p)$ )

Fourth centered moment of a binomial distribution ( $E (W - n p)^{4} = 3 n^{2} q^{2} + n q (1 - 6 q), q = p (1 - p)$ )

Sample proportion ( $p = D / n$ )

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