The Bernoulli variable

Consider a random experiment in which there are only two possible outcomes. Call these outcomes success and failure. Such an experiment is called a Bernoulli trial. Let X be a random variable which takes the value 1 when the outcome is a success and 0 when the outcome is a failure. It is called a Bernoulli variable.

Examples

1. Whether a particular occupant will die in a fire.

2. Whether or not flashover will occur in a compartment fire.

3. Whether a particular sprinkler will operate or not.

Let the probability of success be denoted by p and the probability of failure by q = 1 ? p.

Clearly we have E( X) = 1 p + 0 (1 ? p) = p. Moreover E( X ²) = 1 ² p + 0 ² (1 ? p) = p. It follows that

The binomial distribution

Consider now ? independent repetitions of the experiment ( ? independent trials ) and define the random variable X as the number of successes in ? independent trials.

Clearly, X can assume the values 0, 1, , ?. The probability of x successes (and therefore ? ? x failures) is given by the formula:

where , the binomial coefficient is given...