Binomial Experiment

The binomial distribution is a discrete distribution which implies its sample space is finite. In other words, the values that a binomial random variable can take on is limited. The mean and variance is given by $ \mu = np $ and $ \sigma^2 = npq,$ respectively.

In order to analyze an experiment as a binomial experiment, five conditions must hold:

  1. Only two possible outcomes can occur, success or failure.
  2. The experiment is repeated $n$ times, i.e. there are $ n $ trials.
  3. $p$, the probability of success is constant from trial to trial.
  4. The trials are independent, i.e. the outcome of one trial does not affect another.
  5. The variable of interest is the number of successes.

If these five conditions hold, it is a binomial experiment.

For an example, consider estimating the number of registered voters in Springfield. We can take a random sample and use the sample to estimate the population. A binomial random variable can be used to model the data. Suppose in a sample of size 30, 14 residents of Springfield were registered. We can use the numbers 0 and 1 to signify non-registered and registered voters, respectively. $ \text{P}(X=0) = \frac{16}{30} = 0.533\,\,\, $ and $ \text{P}(X = 1) = \frac{14}{30}=0.467.\,\,\, $ In this example, $ n = 30. \,\, p $ is approximately 0.467 since we are estimating the number of registered voters. $ \bar{x}, $ the sample mean, is equal to 0.467, and $ s^2 = 30(0.467)(0.533) \approx 7.5. $

If we were interested in the number of unregistered voters $ p = 0.533 $ . Usually, we assign $ p $ to the variable of interest. However, this would not have changed the sample variance.