Lesson 4

Two examples of standard and frequently occurring distributions:

• The Bernoulli distribution  with parameter \(p\). The experiment is a single trial with two possible outcomes: success and failure. The random variable \(\alpha\) is equal to one if success occurs and zero otherwise: \[d(1)=p,\quad d(0)=1-p \ .\] It is written as \(\alpha \sim \operatorname{Bernoulli}(p)\).


• The Binomial distribution  with parameters \(n, p\). The experiment is \(n\) independent trials, each ending in success with probability \(p\). The random variable is equal to the number of successes: \[d(k)=\binom{n}{k}p^{k}(1-p)^{n-k}, \text{ for \(0 \le k \le n\).}\] It is written as \(\alpha \sim \operatorname{Binomial}(n,p)\).