The main difference between Bernoulli and Binomial distributions lies in the number of trials and the outcomes they represent.
- Bernoulli Distribution: This distribution deals with the outcome of a single trial of an event, with two possible outcomes: 0 or 1. It is used when the outcome of an event is required for only one time. For example, if you toss a coin with a 25% probability of heads, a single toss of the coin has a Bernoulli distribution, with p = 0.25.
- Binomial Distribution: This distribution deals with the outcome of multiple trials of a single event, where each trial has two possible outcomes: 0 or 1. It is used when the outcome of an event is required multiple times. In order for a random variable to follow a Binomial distribution, the probability of "success" in each Bernoulli trial must be equal and independent. For example, if you toss a coin 3 times and record the number of heads, this is now a Binomial distribution with p = 0.5 (probability of success on a given trial) and n = 3 (number of trials).
In summary:
- Bernoulli distribution represents the outcome of a single trial with two possible outcomes.
- Binomial distribution represents the outcome of multiple trials with two possible outcomes, where the probability of success in each trial is equal and independent.
Comparative Table: Bernoulli vs Binomial
Here is a table comparing the differences between Bernoulli and Binomial distributions:
Feature | Bernoulli Distribution | Binomial Distribution |
---|---|---|
Outcomes | Two possible outcomes: 0 or 1 | Multiple possible outcomes: 0, 1, 2, …, n |
Trials | Single trial | Multiple trials |
Success Probability | Set probability of success on a given trial | Probability of success on each trial is equal and independent |
Applications | Coin flip with fixed probability of heads | Coin flip experiment with multiple trials and different numbers of heads |
In summary, a Bernoulli distribution represents the outcome of a single trial with two possible outcomes, while a Binomial distribution represents the outcomes of multiple trials, each with the same probability of success, and multiple possible outcomes.
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