Common Probability Distribution - Binomial

Binomial Distribution

Definition$

The binomial distribution describes the number of successes ($k$) in a fixed number of independent trials ($n$), each with the same probability of success ($p$). It is a discrete function of integer values (0, 1, 2, …, n) and applicable for binary outcome.

Mathematical Formula

Probability Mass Function

\(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\)

Where:

• $n$ = number of trials
• $k$ = number of successes
• $p$ = probability of success
• $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ (binomial coefficient)

Mean and Variance

\(\mu = np\)

\[\sigma^2 = np(1-p)\]

Python Implementation

For detailed Python implementations, examples, and data science applications, see the accompanying Jupyter notebook:

📓 Binomial Distribution - Python Implementation

Notes

• Normal Approximation: When $n$ is large and $p$ is not sall (one practical choice can be $n>30$ and $p > 0.05$), binomial can be approximated by normal distribution
• Poisson Approximation: When $n$ is large, $p$ is small, and $np$ is finite, binomial approaches Poisson distribution