e-Statistics > 4470-5470 Probability and Statistics I

Binomial distribution

Tossing a coin is an example of independent trial where head (red face) or tail (blue face) are the two possible outcomes. When it is tossed repeatedly, the experiment yields the number of successes $ Z =$ (red faces).

When an experiment involves a ``Bernoulli trial'' such that

  1. it results in possibly two outcomes, say ``success'' or ``failure,'' and
  2. it is repeatedly and independently performed (that is, each trial is designed to be identical but does not interfere with others),
it is called a binomial experiment. The probability of success remains the same probability $ p =$ in each trial, and a trial is performed repeatedly $ n=$ times. (Here the experiment is limited to $ n \le 50$.)

In the experiment the sum $ Z$ of successes is the random variable of interest. The exact frequency function $ f(k) = P(Z = k)$ is formulated as

$\displaystyle f(k) = \binom{n}{k} p^k (1-p)^{n-k}$    for $ k = 0,1,\ldots,n$,

and it is called a binomial distribution with parameters $ n$ and $ p$.