Bernoulli Distribution

Definition

The Bernoulli distribution is a discrete probability distribution that models a random experiment with only two possible outcomes: success or failure. This distribution is fundamental in statistics because it forms the basis for other important distributions, like the binomial distribution, which considers multiple independent Bernoulli trials. It is characterized by a single parameter, usually denoted as 'p', which represents the probability of success in a single trial.

5 Must Know Facts For Your Next Test

1. The probability mass function (PMF) of a Bernoulli distribution can be expressed as $$P(X=1) = p$$ for success and $$P(X=0) = 1 - p$$ for failure.
2. The expected value or mean of a Bernoulli random variable is equal to the probability of success, $$E(X) = p$$.
3. The variance of a Bernoulli random variable is calculated using the formula $$Var(X) = p(1 - p)$$.
4. The Bernoulli distribution is a special case of the binomial distribution where n equals 1, meaning there is only one trial.
5. In practical applications, the Bernoulli distribution can model scenarios such as flipping a coin, where heads represents success and tails represents failure.

Review Questions

• How does the Bernoulli distribution lay the groundwork for understanding more complex distributions like the binomial distribution?
  • The Bernoulli distribution serves as the building block for more complex probability models. In essence, it deals with individual trials where each trial results in either success or failure. The binomial distribution then extends this concept by aggregating multiple independent Bernoulli trials. Thus, understanding the characteristics and behaviors of a single Bernoulli trial enables deeper insights into how repeated trials can combine to form a binomial distribution.
• Compare and contrast the probabilities associated with success and failure in a Bernoulli distribution. What do these probabilities indicate about the outcomes?
  • In a Bernoulli distribution, the probability of success is denoted as 'p', while the probability of failure is represented as '1 - p'. This relationship directly shows that both outcomes are mutually exclusive and exhaustive since they encompass all possible results of a single trial. If 'p' is high, it indicates that success is likely; conversely, if 'p' is low, failure becomes more probable. This comparison highlights how understanding these probabilities can influence decision-making and predictions based on potential outcomes.
• Evaluate how varying the parameter 'p' in the Bernoulli distribution affects its mean and variance. What implications does this have for statistical modeling?
  • When adjusting the parameter 'p' in a Bernoulli distribution, both the mean and variance are impacted. Specifically, as 'p' increases toward 1, the mean approaches 1, indicating a higher likelihood of success. Simultaneously, variance, calculated as $$Var(X) = p(1 - p)$$, decreases because it reflects reduced uncertainty when successes become more frequent. Understanding this relationship is crucial in statistical modeling, as it allows analysts to predict outcomes more accurately based on varying probabilities.

Related terms

Success: The outcome of a Bernoulli trial that corresponds to the event of interest occurring, typically denoted as 1 in the context of the Bernoulli distribution.

Failure: The outcome of a Bernoulli trial that corresponds to the event of interest not occurring, typically denoted as 0 in the context of the Bernoulli distribution.

Binomial Distribution:

A probability distribution that models the number of successes in a fixed number of independent Bernoulli trials, characterized by parameters n (number of trials) and p (probability of success).