Bernoulli trial

A Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted.

Definition

• $X^{B}$ is 1 if outcome is success, or 0 is failure. (Example: Flip a coin, 1=Heads, 0=Tails)
• $P(X^{B}=1)=p$
• $P(X^{B}=0)=1-p=q$

Properties

The sum of n independent Bernouilli random variables $X_{i}^{B}$ with the same parameter p is a binomial random variable $X^{B}$

• $X^{b}=\sum _{i=0}^{n}X_{i}^{B}$
• $P(X^{b}=x)={\frac {n!}{x!(n-x)!}}p^{x}(1-p)^{n-x}={\binom {n}{x}}p^{x}q^{n-x}$