Binomial Distribution
Understanding Binomial Distribution
- A binomial distribution is a specific type of discrete probability distribution.
- It requires the probability experiment to be repeated a fixed number of times, with each trial being independent of the others.
- It also states that there must be only two possible outcomes in a trial- success or failure- the probability of success remaining constant throughout.
- Each event in a binomial experiment is independent of the others, which means the outcome of one event does not affect the outcome of any other event.
Features and Parameters of Binomial Distribution
- There are two parameters in a binomial distribution: the number of trials (n) and the probability of success in a single trial (p).
- The mean of a binomial distribution is np, and the standard deviation is sqrt(npq), where q=1-p.
- Binomial probabilities are calculated using the binomial formula: P(X=k) = nCk *pk * (1-p)n-k, where nCk represents the number of combinations of n trials taken k at a time.
Implementing Binomial Distribution
- In practical scenarios, binomial distribution is used when one is interested in the number of successes k in a fixed number of trials n for a binary process where the chance of success in each trial is the same, p.
- It can also be used to conduct hypothesis testing and to build confidence intervals around sample estimates.
Assumptions of Binomial Distribution
- The outcomes of each trial are only of two categories, a success or a failure.
- The probability of a success is the same in each trial.
- The trials are independent, implying the outcome of one trial does not have any effect on the outcome of other trials.
- The number of trials, n, is fixed in advance.
Errors and Uncertainty in Binomial Distribution
- A binomial distribution can be a poor model if the probability of success is not the same in each trial or if the trials are not independent.
- The spread or the variability of a binomial distribution increases as the probability p approaches 0.5. Hence, a binomial experiment with p near 0.5 will have more spread as compared to an experiment with p near 0 or 1.