| Binomial distribution | CDF | I q ( n − ⌊ k ⌋ , 1 + ⌊ k ⌋ ) {\displaystyle I_{q}(n-\lfloor k\rfloor ,1+\lfloor k\rfloor )} (the regularized incomplete beta function) | 1.00 | infobox |
| Binomial distribution | CF | ( q + p e i t ) n {\displaystyle (q+pe^{it})^{n}} | 1.00 | infobox |
| Binomial distribution | Entropy | 1 2 log 2 ( 2 π e n p q ) + O ( 1 n ) {\displaystyle {\frac {1}{2}}\log _{2}(2\pi enpq)+O\left({\frac {1}{n}}\right)} in shannons. For nats, use the natural log in the log. | 1.00 | infobox |
| Binomial distribution | Excess kurtosis | 1 − 6 p q n p q {\displaystyle {\frac {1-6pq}{npq}}} | 1.00 | infobox |
| Binomial distribution | Fisher information | g n ( p ) = n p q {\displaystyle g_{n}(p)={\frac {n}{pq}}} (for fixed n {\displaystyle n} ) | 1.00 | infobox |
| Binomial distribution | Mean | n p {\displaystyle np} | 1.00 | infobox |
| Binomial distribution | Median | ⌊ n p ⌋ {\displaystyle \lfloor np\rfloor } or ⌈ n p ⌉ {\displaystyle \lceil np\rceil } | 1.00 | infobox |
| Binomial distribution | MGF | ( q + p e t ) n {\displaystyle (q+pe^{t})^{n}} | 1.00 | infobox |
| Binomial distribution | Mode | ⌊ ( n + 1 ) p ⌋ {\displaystyle \lfloor (n+1)p\rfloor } or ⌈ ( n + 1 ) p ⌉ − 1 {\displaystyle \lceil (n+1)p\rceil -1} | 1.00 | infobox |
| Binomial distribution | Notation | B ( n , p ) {\displaystyle \operatorname {B} (n,p)} | 1.00 | infobox |
| Binomial distribution | Parameters | n ∈ { 0 , 1 , 2 , … } {\displaystyle n\in \{0,1,2,\ldots \}} – number of trials p ∈ [ 0 , 1 ] {\displaystyle p\in [0,1]} – success probability for each trial q = 1 − p {\display… | 1.00 | infobox |
| Binomial distribution | PGF | G ( z ) = [ q + p z ] n {\displaystyle G(z)=[q+pz]^{n}} | 1.00 | infobox |
| Binomial distribution | PMF | ( n k ) p k q n − k {\displaystyle {\binom {n}{k}}p^{k}q^{n-k}} | 1.00 | infobox |
| Binomial distribution | Skewness | q − p n p q {\displaystyle {\frac {q-p}{\sqrt {npq}}}} | 1.00 | infobox |
| Binomial distribution | Support | k ∈ { 0 , 1 , … , n } {\displaystyle k\in \{0,1,\ldots ,n\}} – number of successes | 1.00 | infobox |
| Binomial distribution | Variance | n p q = n p ( 1 − p ) {\displaystyle npq=np(1-p)} | 1.00 | infobox |
| Binomial distribution | is a | Bernoulli distribution | 0.90 | text |
| Binomial distribution | is a | basis for the binomial test of statistical significance.The binomial distribution is frequently used to model the number of successes in a sample of size n drawn with replacemen… | 0.90 | text |
| Binomial distribution | is a | special case of the Poisson binomial distribution | 0.90 | text |
| Binomial distribution | is a | PMF of k successes given n independent events each with a probability p of success | 0.90 | text |
