Bernoulli Trials and Binomial Distribution A Bernoulli trial is a single experiment that can result in only two possible outcomes, often called success and...
Bernoulli Trials and Binomial Distribution
A Bernoulli trial is a single experiment that can result in only two possible outcomes, often called success and failure. The probability of success is constant throughout the trial, and the probability of failure is equal to the complement of the success probability.
The binomial distribution is a probability distribution that describes the number of successes in a sequence of independent Bernoulli trials, where the probability of success is constant. It is often used to model the number of heads or tails in a sequence of coin flips or rolls.
The probability mass function of the binomial distribution is given by:
where:
(n) is the total number of trials.
(k) is the number of successes.
(p) is the probability of success on each trial.
The expected value of a Bernoulli random variable (X) with parameter (p) is given by:
The variance of a Bernoulli random variable (X) with parameter (p) is given by:
The binomial distribution is a versatile distribution that can be used to model a wide variety of real-world phenomena, including the number of defective items in a batch of products, the number of customers in a queue, and the number of heads or tails in a sequence of coin flips