Binomial Distribution

This is a distribution of a series of bernoulli distribution tests. It models the number of successes in a fixed number of independent, identical trials.

1. The Core Idea

To use a Binomial distribution, these Constraints are required. They are intuitive really.

• Binary: Each trial has only two outcomes (Success or Failure).
• Independent: The outcome of one trial doesn't affect the next.
• Number: The total number of trials ($n$) is fixed in advance.
• Same probability: The probability of success ($p$) stays constant for every trial.

We denote a random variable $X$ following a Binomial distribution as:

$$X \sim \text{Binomial}(n, p)$$

Where:

• $n$ = total number of trials
• $p$ = probability of success on a single trial
• $X$ = total number of successes (can be any integer from $0$ to $n$)

Note: X is the number of successes only. If you wanna model something like probability of "HHT" appear in a 10 times coin flip experiment, binomial is NOT the mode. This is a case of Markov Chains or Pattern Probability

PMF

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

• Underlined part is the Bernoulli Distribution

Properties

Mean (Expected Value)

$$E[X] = np$$

Variance

$$\text{Var}(X) = np(1 - p)$$