Factsheet: Binomial distribution
Where to use: The binomial distribution is used when there are a fixed number of trials (\(n\)) and only two possible outcomes for each trial, representing \(n\) many Bernoulli trials. Here, the random variable \(X\) represents the number of successes.
Notation: \(X \sim \textrm{Binomial}(n,p)\) or \(X \sim B(n,p)\).
Parameters: Two numbers \(n,p\) where: - \(n\) is an integer representing the number of trials, - \(p\) is a real number representing the probability of success of a trial (where \(0 \le p \le 1\)).
| Quantity | Value | Notes |
|---|---|---|
| Mean | \(\mathbb{E}(X) = np\) | |
| Variance | \(\mathbb{V}(X) = np(1-p)\) | |
| PMF | \(\mathbb{P}(X=x)=\dfrac{n!}{(n-x)!x!}p^xq^{(n-x)}\) | |
| CDF | \(\mathbb{P}(X\leq x)=I_{q}(n-\lfloor x \rfloor,1+\lfloor x \rfloor)\) | \(I_x(a,b)\) regularized incomplete beta function, \(\lfloor x \rfloor\) the floor function |
Example: You flip a coin \(10\) times, and the probability of getting ‘heads’ is \(0.5\). Taking ‘heads’ as a success, this can be expressed as \(X \sim B(10, 0.5)\), meaning \(10\) trials are conducted, where the probability of success in each trial is \(0.5\).
Further reading
This interactive element appears in Guide: PMFs, PDFs, CDFs and Overview: Probability distributions. Please click the relevant links to go to the guides.
Version history
v1.0: initial version created 04/25 by tdhc and Michelle Arnetta as part of a University of St Andrews VIP project.
v1.1: moved to factsheet form and populated with material from Overview: Probability distributions by tdhc.
v1.2: deprecated shinylive apps replaced 09/26 by tdhc.