Binomial Distribution

Statistics Econometrics
The binomial distribution gives the probability of getting a fixed number of successes in a fixed number of independent trials.
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The binomial distribution describes the probability of observing exactly a given number of successes when there are n independent trials and the probability of success is the same in each trial.

The basic formula

If X is the number of successes, then:

P(X = k) = C(n,k) p^k (1-p)^(n-k)

where:

  • n is the number of trials,
  • k is the number of successes,
  • p is the probability of success on each trial.

Its mean is np, and its variance is np(1-p).

When it applies

The binomial distribution is appropriate only when all of the following hold:

  • the number of trials is fixed,
  • each trial has two outcomes,
  • trials are independent,
  • the success probability is constant.

If those assumptions break down, economists usually need a different model.

Why it matters in economics

Economists use binomial logic when counting events such as:

  • the number of survey respondents who support a policy,
  • the number of borrowers who default in a small homogeneous pool,
  • the number of firms that adopt a policy under a simple yes-or-no setup.

The distribution is also a building block for confidence intervals, hypothesis testing, and introductory discrete-outcome modeling.

Knowledge Check

### Which condition is required for a binomial distribution? - [x] Each trial has the same probability of success - [ ] The variable must be continuous - [ ] The number of trials must be random - [ ] The outcomes must have more than two categories > **Explanation:** A binomial setup requires identical success probability across independent binary trials. ### What is the mean of a binomial random variable with parameters `n` and `p`? - [x] `np` - [ ] `n + p` - [ ] `p / n` - [ ] `n(1-p)` > **Explanation:** The expected number of successes is the number of trials times the success probability. ### Why might an economist reject a binomial model for defaults in a loan portfolio? - [x] Because defaults may be correlated rather than independent - [ ] Because default is not a binary event - [ ] Because probabilities must always equal 0.5 - [ ] Because binomial models cannot count outcomes > **Explanation:** If defaults move together during a downturn, the independence assumption behind the binomial distribution becomes weak.
Binomial Pricing
Binary Choice Models