## 2.4 Bernoulli Sequence and Binomial Distribution

#### The Bernoulli Sequence

- There are many types of random variables that contributes to a discrete probability distribution. For instance, the number of ‘sixes’ obtained from 10 throws of a fair dice, and the amount of lottery winning are examples of a discrete random variables.

- For a trial that has only either success or fail with a fixed probability, we call this as a Bernoulli trial. That is, the probability distribution table is given by

x | \text { Outcome } 1 \text { (Success) } | \text { Outcome } 2 \text { (Fail) } |

p(x) | p | 1-p |

where p is a constant.

- Following this logic, we can also say that Bernoulli random variables are used to model two-outcome situations with fixed probability for each outcome.

- In Bernoulli trial, we define success as x=1, and fail as x=0. Therefore, we can calculate the mean of a Bernoulli trial random variable as p, and its variance as p(1-p).

- A particular expansion of this is the Bernoulli sequence. It describes a sequence of repeated trials with the following properties: