Binomial Distribution

The Binomial distribution measures the number of successes in n tests of Bernoulli equal and independent

The discrete v.a. \xi we measured the number of successes in n tests of Bernoulli equal and independent is said to follow a binomial distribution of parameters n and p = P\{success\} and is denoted how to:

\xi \approx B(n, p)

Its probability function is:

P\{\xi = k \} = \binom{k}{n} \cdot p^k \cdot q^{n - k}, k \in \{0, \cdots, n\}

E(\xi) = n \cdot p

\sigma^2(\xi) = n \cdot p \cdot q

\sigma(\xi) = +\sqrt{n \cdot p \cdot q}

Properties of the Binomial distribution

1. \xi = \xi_1 + \xi_2 \approx B(n_1 + n_2, p) when \xi_1, \xi_2 are v.a. independent
2. \xi = \xi_1 + \cdots + \xi_r \approx B(n_1 + \cdots + r_n, p) when xi_1, \cdots, \xi_r are v.a. independent

Calculation of a Binomial

n p k

\xi \approx B(n, p)

E(\xi)
\sigma^2(\xi)
\sigma(\xi)