Bernoulli Distribution

Definition

The Bernoulli distribution is a discrete probability distribution that describes a random experiment with exactly two outcomes: success or failure, usually coded as 1 and 0, respectively. This distribution is foundational in probability theory, as it serves as the basis for more complex distributions like the binomial distribution. The Bernoulli distribution is characterized by its parameter 'p', which represents the probability of success on a single trial.

5 Must Know Facts For Your Next Test

1. The Bernoulli distribution has only one parameter, 'p', which indicates the probability of success; thus, the probability of failure is '1 - p'.
2. The expected value (mean) of a Bernoulli random variable is equal to 'p', while the variance is equal to 'p(1 - p)'.
3. If you perform multiple Bernoulli trials, the total number of successes follows a binomial distribution, which sums up several independent Bernoulli trials.
4. The Bernoulli distribution can model various real-life scenarios, such as flipping a coin (heads or tails) or determining if a product is defective (defective or not defective).
5. In cases where 'p' equals 0.5, the Bernoulli distribution represents a fair game situation, meaning that the chances of success and failure are equally likely.

Review Questions

• How does the Bernoulli distribution serve as a building block for understanding other discrete distributions?
  • The Bernoulli distribution is crucial because it models a single trial with two possible outcomes. When you have multiple independent trials, the outcomes can be aggregated using the Bernoulli results to form other distributions like the binomial distribution. This connection shows how understanding individual Bernoulli trials helps in analyzing scenarios involving multiple attempts or experiments.
• What are the implications of the expected value and variance in a Bernoulli distribution when interpreting its behavior?
  • The expected value gives insight into what we can anticipate as an average outcome over many trials, equating to 'p' for a Bernoulli random variable. The variance reflects how much variability there is around that average outcome, calculated as 'p(1 - p)', indicating that when 'p' is 0.5, variability is at its highest. Understanding both measures helps in assessing risk and making predictions about future outcomes in experiments modeled by this distribution.
• Evaluate how variations in the probability parameter 'p' influence real-world situations modeled by a Bernoulli distribution.
  • Variations in 'p' dramatically impact the interpretation of results from Bernoulli trials in real-world situations. For example, if 'p' represents customer satisfaction in a survey, increasing 'p' indicates better overall satisfaction rates. Conversely, a lower 'p' may suggest issues that need addressing. Thus, analyzing different values of 'p' enables organizations to make informed decisions based on probabilities derived from actual data.

Related terms

Binomial Distribution:

A probability distribution that represents the number of successes in a fixed number of independent Bernoulli trials, with each trial having the same probability of success.

Expected Value:

The long-term average value of a random variable, calculated as the sum of all possible values weighted by their probabilities.

Variance:

A measure of the dispersion of a set of values, calculated as the expected value of the squared deviation of each value from the mean.