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) }

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:
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