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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} $
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